Pascal matrix generation
You are encouraged to solve this task according to the task description, using any language you may know.
A pascal matrix is a two-dimensional square matrix holding numbers from Pascal's triangle, also known as binomial coefficients and which can be shown as nCr.
Shown below are truncated 5-by-5 matrices M[i, j] for i,j in range 0..4.
A Pascal upper-triangular matrix that is populated with jCi:
[[1, 1, 1, 1, 1], [0, 1, 2, 3, 4], [0, 0, 1, 3, 6], [0, 0, 0, 1, 4], [0, 0, 0, 0, 1]]
A Pascal lower-triangular matrix that is populated with iCj (the transpose of the upper-triangular matrix):
[[1, 0, 0, 0, 0], [1, 1, 0, 0, 0], [1, 2, 1, 0, 0], [1, 3, 3, 1, 0], [1, 4, 6, 4, 1]]
A Pascal symmetric matrix that is populated with i+jCi:
[[1, 1, 1, 1, 1], [1, 2, 3, 4, 5], [1, 3, 6, 10, 15], [1, 4, 10, 20, 35], [1, 5, 15, 35, 70]]
- Task
Write functions capable of generating each of the three forms of n-by-n matrices.
Use those functions to display upper, lower, and symmetric Pascal 5-by-5 matrices on this page.
The output should distinguish between different matrices and the rows of each matrix (no showing a list of 25 numbers assuming the reader should split it into rows).
- Note
The Cholesky decomposition of a Pascal symmetric matrix is the Pascal lower-triangle matrix of the same size.
11l
<lang 11l>F pascal_upp(n)
V s = [[0] * n] * n
s[0] = [1] * n
L(i) 1 .< n
L(j) i .< n
s[i][j] = s[i - 1][j - 1] + s[i][j - 1]
R s
F pascal_low(n)
V upp = pascal_upp(n)
V s = [[0] * n] * n
L(x) 0 .< n
L(y) 0 .< n
s[y][x] = upp[x][y]
R s
F pascal_sym(n)
V s = [[1] * n] * n
L(i) 1 .< n
L(j) 1 .< n
s[i][j] = s[i - 1][j] + s[i][j - 1]
R s
F pp(mat)
print(‘[’mat.map(String).join(",\n ")‘]’)
-V n = 5 print(‘Upper:’) pp(pascal_upp(n)) print("\nLower:") pp(pascal_low(n)) print("\nSymmetric:") pp(pascal_sym(n))</lang>
- Output:
Upper: [[1, 1, 1, 1, 1], [0, 1, 2, 3, 4], [0, 0, 1, 3, 6], [0, 0, 0, 1, 4], [0, 0, 0, 0, 1]] Lower: [[1, 0, 0, 0, 0], [1, 1, 0, 0, 0], [1, 2, 1, 0, 0], [1, 3, 3, 1, 0], [1, 4, 6, 4, 1]] Symmetric: [[1, 1, 1, 1, 1], [1, 2, 3, 4, 5], [1, 3, 6, 10, 15], [1, 4, 10, 20, 35], [1, 5, 15, 35, 70]]
360 Assembly
<lang 360asm>* Pascal matrix generation - 10/06/2018 PASCMATR CSECT
USING PASCMATR,R13 base register
B 72(R15) skip savearea
DC 17F'0' savearea
SAVE (14,12) save previous context
ST R13,4(R15) link backward
ST R15,8(R13) link forward
LR R13,R15 set addressability
MVC MAT,=F'1' mat(1,1)=1
LA R6,1 i=1
DO WHILE=(C,R6,LE,N) do i=1 to n;
LA R7,1 j=1
DO WHILE=(C,R7,LE,N) do j=1 to n;
LR R2,R6 i
LA R3,1(R7) r3=j+1
LR R1,R6 i
BCTR R1,0 -1
MH R1,NN *nn
AR R1,R7 ~(i,j)
SLA R1,2 *4
L R4,MAT-4(R1) r4=mat(i,j)
LR R5,R6 i
MH R5,NN *nn
AR R5,R7 ~(i+1,j)
SLA R5,2 *4
L R5,MAT-4(R5) r5=mat(i+1,j)
AR R4,R5 r4=mat(i,j)+mat(i+1,j)
MH R2,NN *nn
AR R2,R3 ~(i+1,j+1)
SLA R2,2 *4
ST R4,MAT-4(R2) mat(i+1,j+1)=mat(i,j)+mat(i+1,j)
LA R7,1(R7) j++
ENDDO , enddo j
LA R6,1(R6) i++
ENDDO , enddo i
MVC TITLE,=CL20'Upper:'
BAL R14,PRINTMAT call printmat
MVC MAT,=F'1' mat(1,1)=1
LA R6,1 i=1
DO WHILE=(C,R6,LE,N) do i=1 to n;
LA R7,1 j=1
DO WHILE=(C,R7,LE,N) do j=1 to n;
LR R2,R6 i
LA R3,1(R7) r3=j+1
LR R1,R6 i
BCTR R1,0 -1
MH R1,NN *nn
LR R0,R7 j
AR R1,R0 ~(i,j)
SLA R1,2 *4
L R4,MAT-4(R1) r4=mat(i,j)
LA R5,1(R7) j+1
LR R1,R6 i
BCTR R1,0 -1
MH R1,NN *nn
AR R1,R5 ~(i,j+1)
SLA R1,2 *4
L R5,MAT-4(R1) r5=mat(i,j+1)
AR R4,R5 mat(i,j)+mat(i,j+1)
MH R2,NN *nn
AR R2,R3 ~(i+1,j+1)
SLA R2,2 *4
ST R4,MAT-4(R2) mat(i+1,j+1)=mat(i,j)+mat(i,j+1)
LA R7,1(R7) j++
ENDDO , enddo j
LA R6,1(R6) i++
ENDDO , enddo i
MVC TITLE,=CL20'Lower:'
BAL R14,PRINTMAT call printmat
MVC MAT+24,=F'1' mat(2,1)=1
LA R6,1 i=1
DO WHILE=(C,R6,LE,N) do i=1 to n;
LA R7,1 j=1
DO WHILE=(C,R7,LE,N) do j=1 to n;
LR R2,R6 i
LA R3,1(R7) r3=j+1 j
LR R1,R6 i
BCTR R1,0 -1
MH R1,NN *nn
AR R1,R3 ~(i,j+1)
SLA R1,2 *4
L R4,MAT-4(R1) r4=mat(i,j+1)
LR R5,R6 i
MH R5,NN *nn
AR R5,R7 j
SLA R5,2 *4
L R5,MAT-4(R5) r5=mat(i+1,j)
AR R4,R5 mat(i,j+1)+mat(i+1,j)
MH R2,NN *nn
AR R2,R3 ~(i+1,j+1)
SLA R2,2 *4
ST R4,MAT-4(R2) mat(i+1,j+1)=mat(i,j+1)+mat(i+1,j)
LA R7,1(R7) j++
ENDDO , enddo j
LA R6,1(R6) i++
ENDDO , enddo i
MVC TITLE,=CL20'Symmetric:'
BAL R14,PRINTMAT call printmat
L R13,4(0,R13) restore previous savearea pointer
RETURN (14,12),RC=0 restore registers from calling sav
PRINTMAT XPRNT TITLE,L'TITLE print title -----------------------
LA R10,PG pgi=0
LA R6,1 i=1
DO WHILE=(C,R6,LE,N) do i=1 to n;
LA R7,1 j=1
DO WHILE=(C,R7,LE,N) do j=1 to n;
LR R2,R6 i
LR R3,R7 j
LA R3,1(R3) j+1
MH R2,NN *nn
AR R2,R3 ~(i+1,j+1)
SLA R2,2 *4
L R2,MAT-4(R2) mat(i+1,j+1)
XDECO R2,XDEC edit mat(i+1,j+1)
MVC 0(5,R10),XDEC+7 output mat(i+1,j+1)
LA R10,5(R10) pgi+=5
LA R7,1(R7) j++
ENDDO , enddo j
XPRNT PG,L'PG print
LA R10,PG pgi=0
LA R6,1(R6) i++
ENDDO , enddo i
BR R14 return to caller -------------------
X EQU 5 matrix size N DC A(X) n=x NN DC AL2(X+1) nn=x+1 MAT DC ((X+1)*(X+1))F'0' mat(x+1,x+1) TITLE DC CL20' ' title PG DC CL80' ' buffer PGI DC H'0' buffer index XDEC DS CL12 temp
YREGS
END PASCMATR</lang>
- Output:
Upper:
1 1 1 1 1
0 1 2 3 4
0 0 1 3 6
0 0 0 1 4
0 0 0 0 1
Lower:
1 0 0 0 0
1 1 0 0 0
1 2 1 0 0
1 3 3 1 0
1 4 6 4 1
Symmetric:
1 1 1 1 1
1 2 3 4 5
1 3 6 10 15
1 4 10 20 35
1 5 15 35 70
Action!
<lang Action!>BYTE FUNC Index(BYTE i,j,dim) RETURN (i*dim+j)
PROC PascalUpper(BYTE ARRAY mat BYTE dim)
BYTE i,j
FOR i=0 TO dim-1
DO
FOR j=0 TO dim-1
DO
IF i>j THEN
mat(Index(i,j,dim))=0
ELSEIF i=j OR i=0 THEN
mat(Index(i,j,dim))=1
ELSE
mat(Index(i,j,dim))=mat(Index(i-1,j-1,dim))+mat(Index(i,j-1,dim))
FI
OD
OD
RETURN
PROC PascalLower(BYTE ARRAY mat BYTE dim)
BYTE i,j
FOR i=0 TO dim-1
DO
FOR j=0 TO dim-1
DO
IF i<j THEN
mat(Index(i,j,dim))=0
ELSEIF i=j OR j=0 THEN
mat(Index(i,j,dim))=1
ELSE
mat(Index(i,j,dim))=mat(Index(i-1,j-1,dim))+mat(Index(i-1,j,dim))
FI
OD
OD
RETURN
PROC PascalSymmetric(BYTE ARRAY mat BYTE dim)
BYTE i,j
FOR i=0 TO dim-1
DO
FOR j=0 TO dim-1
DO
IF i=0 OR j=0 THEN
mat(Index(i,j,dim))=1
ELSE
mat(Index(i,j,dim))=mat(Index(i-1,j,dim))+mat(Index(i,j-1,dim))
FI
OD
OD
RETURN
PROC PrintMatrix(BYTE ARRAY mat BYTE dim)
BYTE i,j,v
FOR i=0 TO dim-1
DO
FOR j=0 TO dim-1
DO
v=mat(Index(i,j,dim))
IF v<10 THEN
Print(" ")
ELSEIF v<100 THEN
Print(" ")
FI
PrintB(v)
OD
PutE()
OD
RETURN
PROC Main()
BYTE ARRAY mat(25) BYTE dim=[5]
PrintE("Pascal upper matrix:")
PascalUpper(mat,dim)
PrintMatrix(mat,dim)
PutE()
PrintE("Pascal lower matrix:")
PascalLower(mat,dim)
PrintMatrix(mat,dim)
PutE()
PrintE("Pascal symmetric matrix:")
PascalSymmetric(mat,dim)
PrintMatrix(mat,dim)
RETURN</lang>
- Output:
Screenshot from Atari 8-bit computer
Pascal upper matrix: 1 1 1 1 1 0 1 2 3 4 0 0 1 3 6 0 0 0 1 4 0 0 0 0 1 Pascal lower matrix: 1 0 0 0 0 1 1 0 0 0 1 2 1 0 0 1 3 3 1 0 1 4 6 4 1 Pascal symmetric matrix: 1 1 1 1 1 1 2 3 4 5 1 3 6 10 15 1 4 10 20 35 1 5 15 35 70
Ada
<lang ada>-- for I/O with Ada.Text_IO; use Ada.Text_IO; with Ada.Integer_Text_IO; use Ada.Integer_Text_IO; with Ada.Float_Text_IO; use Ada.Float_Text_IO;
-- for estimating the maximum width of a column with Ada.Numerics.Generic_Elementary_Functions;
procedure PascalMatrix is
type Matrix is array (Positive range <>, Positive range <>) of Natural;
-- instantiate Generic_Elementary_Functions for Float type
package Math is new Ada.Numerics.Generic_Elementary_Functions(Float_Type => Float);
use Math;
procedure Print(m: in Matrix) is
-- determine the maximum width of a column
w: Float := Log(Float(m'Length(1)**(m'Length(1)/2)), 10.0);
width: Positive := Natural(Float'Ceiling(w)) + 1;
begin
for i in m'First(1)..m'Last(1) loop
Put("( ");
for j in m'First(2)..m'Last(2) loop
Put(m(i,j), width);
end loop;
Put(" )"); New_Line(1);
end loop;
end Print;
function Upper_Triangular(n: in Positive) return Matrix is
result: Matrix(1..n, 1..n) := (
1 => ( others => 1 ),
others => ( others => 0 )
);
begin
for i in 2..n loop
result(i,i) := 1;
for j in i+1..n loop
result(i,j) := result(i,j-1) + result(i-1,j-1);
end loop;
end loop;
return result;
end Upper_Triangular;
function Lower_Triangular(n: in Positive) return Matrix is
result: Matrix(1..n, 1..n) := (
others => ( 1 => 1, others => 0 )
);
begin
for i in 2..n loop
result(i,i) := 1;
for j in i+1..n loop
result(j,i) := result(j-1,i) + result(j-1,i-1);
end loop;
end loop;
return result;
end Lower_Triangular;
function Symmetric(n: in Positive) return Matrix is
result: Matrix(1..n, 1..n) := (
1 => ( others => 1 ),
others => ( 1 => 1, others => 0 )
);
begin
for i in 2..n loop
for j in 2..n loop
result(i,j) := result(i,j-1) + result(i-1,j);
end loop;
end loop;
return result;
end Symmetric;
n: Positive;
begin
Put("What dimension Pascal matrix would you like? ");
Get(n);
Put("Upper triangular:"); New_Line(1);
Print(Upper_Triangular(n));
Put("Lower triangular:"); New_Line(1);
Print(Lower_Triangular(n));
Put("Symmetric:"); New_Line(1);
Print(Symmetric(n));
end PascalMatrix;</lang>
- Output:
What dimension Pascal matrix would you like? 5 Upper triangular: ( 1 1 1 1 1 ) ( 0 1 2 3 4 ) ( 0 0 1 3 6 ) ( 0 0 0 1 4 ) ( 0 0 0 0 1 ) Lower triangular: ( 1 0 0 0 0 ) ( 1 1 0 0 0 ) ( 1 2 1 0 0 ) ( 1 3 3 1 0 ) ( 1 4 6 4 1 ) Symmetric: ( 1 1 1 1 1 ) ( 1 2 3 4 5 ) ( 1 3 6 10 15 ) ( 1 4 10 20 35 ) ( 1 5 15 35 70 )
ALGOL 68
<lang algol68>BEGIN
# returns an upper Pascal matrix of size n #
PROC upper pascal matrix = ( INT n )[,]INT:
BEGIN
[ 1 : n, 1 : n ]INT result;
FOR j TO n DO result[ 1, j ] := 1 OD;
FOR i FROM 2 TO n DO
result[ i, 1 ] := 0;
FOR j FROM 2 TO n DO
result[ i, j ] := result[ i - 1, j - 1 ] + result[ i, j - 1 ]
OD
OD;
result
END # upper pascal matrix # ;
# returns a lower Pascal matrix of size n #
PROC lower pascal matrix = ( INT n )[,]INT:
BEGIN
[ 1 : n, 1 : n ]INT result;
FOR i TO n DO result[ i, 1 ] := 1 OD;
FOR j FROM 2 TO n DO
result[ 1, j ] := 0;
FOR i FROM 2 TO n DO
result[ i, j ] := result[ i - 1, j - 1 ] + result[ i - 1, j ]
OD
OD;
result
END # lower pascal matrix # ;
# returns a symmetric Pascal matrix of size n #
PROC symmetric pascal matrix = ( INT n )[,]INT:
BEGIN
[ 1 : n, 1 : n ]INT result;
FOR i TO n DO
result[ i, 1 ] := 1;
result[ 1, i ] := 1
OD;
FOR j FROM 2 TO n DO
FOR i FROM 2 TO n DO
result[ i, j ] := result[ i, j - 1 ] + result[ i - 1, j ]
OD
OD;
result
END # symmetric pascal matrix # ;
# print the matrix m with the specified field width #
PROC print matrix = ( [,]INT m, INT field width )VOID:
BEGIN
FOR i FROM 1 LWB m TO 1 UPB m DO
FOR j FROM 2 LWB m TO 2 UPB m DO
print( ( " ", whole( m[ i, j ], - field width ) ) )
OD;
print( ( newline ) )
OD
END # print matrix # ;
print( ( "upper:", newline ) ); print matrix( upper pascal matrix( 5 ), 2 ); print( ( "lower:", newline ) ); print matrix( lower pascal matrix( 5 ), 2 ); print( ( "symmetric:", newline ) ); print matrix( symmetric pascal matrix( 5 ), 2 )
END</lang>
- Output:
upper: 1 1 1 1 1 0 1 2 3 4 0 0 1 3 6 0 0 0 1 4 0 0 0 0 1 lower: 1 0 0 0 0 1 1 0 0 0 1 2 1 0 0 1 3 3 1 0 1 4 6 4 1 symmetric: 1 1 1 1 1 1 2 3 4 5 1 3 6 10 15 1 4 10 20 35 1 5 15 35 70
ALGOL W
<lang algolw>begin
% initialises m to an upper Pascal matrix of size n %
% the bounds of m must be at least 1 :: n, 1 :: n %
procedure upperPascalMatrix ( integer array m( *, * )
; integer value n
) ;
begin
for j := 1 until n do m( 1, j ) := 1;
for i := 2 until n do begin
m( i, 1 ) := 0;
for j := 2 until n do m( i, j ) := m( i - 1, j - 1 ) + m( i, j - 1 )
end for_i
end upperPascalMatrix ;
% initialises m to a lower Pascal matrix of size n %
% the bounds of m must be at least 1 :: n, 1 :: n %
procedure lowerPascalMatrix ( integer array m( *, * )
; integer value n
) ;
begin
for i := 1 until n do m( i, 1 ) := 1;
for j := 2 until n do begin
m( 1, j ) := 0;
for i := 2 until n do m( i, j ) := m( i - 1, j - 1 ) + m( i - 1, j )
end for_j
end lowerPascalMatrix ;
% initialises m to a symmetric Pascal matrix of size n %
% the bounds of m must be at least 1 :: n, 1 :: n %
procedure symmetricPascalMatrix ( integer array m( *, * )
; integer value n
) ;
begin
for i := 1 until n do begin
m( i, 1 ) := 1;
m( 1, i ) := 1
end for_i;
for j := 2 until n do for i := 2 until n do m( i, j ) := m( i, j - 1 ) + m( i - 1, j )
end symmetricPascalMatrix ;
begin % test the pascal matrix procedures %
% print the matrix m with the specified field width %
% the bounds of m must be at least 1 :: n, 1 :: n %
procedure printMatrix ( integer array m( *, * )
; integer value n
; integer value fieldWidth
) ;
begin
for i := 1 until n do begin
write( i_w := fieldWidth, s_w := 0, " ", m( i, 1 ) );
for j := 2 until n do writeon( i_w := fieldWidth, s_w := 0, " ", m( i, j ) )
end for_i
end printMatrix ;
integer array m( 1 :: 10, 1 :: 10 );
integer n, w;
n := 5; w := 2;
upperPascalMatrix( m, n ); write( "upper:" ); printMatrix( m, n, w );
lowerPascalMatrix( m, n ); write( "lower:" ); printMatrix( m, n, w );
symmetricPascalMatrix( m, n ); write( "symmetric:" ); printMatrix( m, n, w )
end
end.</lang>
- Output:
upper: 1 1 1 1 1 0 1 2 3 4 0 0 1 3 6 0 0 0 1 4 0 0 0 0 1 lower: 1 0 0 0 0 1 1 0 0 0 1 2 1 0 0 1 3 3 1 0 1 4 6 4 1 symmetric: 1 1 1 1 1 1 2 3 4 5 1 3 6 10 15 1 4 10 20 35 1 5 15 35 70
APL
<lang APL>upper ← ∘.!⍨¯1+⍳ lower ← ⍉(∘.!⍨¯1+⍳) symmetric ← (⊢![1]∘.+⍨)¯1+⍳</lang>
- Output:
((⊂upper),(⊂lower),(⊂symmetric)) 5 ┌─────────┬─────────┬────────────┐ │1 1 1 1 1│1 0 0 0 0│1 1 1 1 1│ │0 1 2 3 4│1 1 0 0 0│1 2 3 4 5│ │0 0 1 3 6│1 2 1 0 0│1 3 6 10 15│ │0 0 0 1 4│1 3 3 1 0│1 4 10 20 35│ │0 0 0 0 1│1 4 6 4 1│1 5 15 35 70│ └─────────┴─────────┴────────────┘
AppleScript
By composition of generic functions: <lang AppleScript>-- PASCAL MATRIX -------------------------------------------------------------
-- pascalMatrix :: ((Int, Int) -> (Int, Int)) -> Int -> Int on pascalMatrix(f, n)
chunksOf(n, map(compose(my bc, f), range({{0, 0}, {n - 1, n - 1}})))
end pascalMatrix
-- Binomial coefficient -- bc :: (Int, Int) -> Int on bc(nk)
set {n, k} to nk
script bc_
on |λ|(a, x)
floor((a * (n - x + 1)) / x)
end |λ|
end script
foldl(bc_, 1, enumFromTo(1, k))
end bc
-- TEST ----------------------------------------------------------------------
on run
set matrixSize to 5
script symm
on |λ|(ab)
set {a, b} to ab
{a + b, a}
end |λ|
end script
script format
on |λ|(s, xs)
unlines(concat({{s}, map(my show, xs), {""}}))
end |λ|
end script
unlines(zipWith(format, ¬
{"Lower", "Upper", "Symmetric"}, ¬
|<*>|(map(curry(pascalMatrix), [|id|, swap, symm]), {matrixSize})))
end run
-- GENERIC FUNCTIONS ---------------------------------------------------------
-- A list of functions applied to a list of arguments -- (<*> | ap) :: [(a -> b)] -> [a] -> [b] on |<*>|(fs, xs)
set {nf, nx} to {length of fs, length of xs}
set acc to {}
repeat with i from 1 to nf
tell mReturn(item i of fs)
repeat with j from 1 to nx
set end of acc to |λ|(contents of (item j of xs))
end repeat
end tell
end repeat
return acc
end |<*>|
-- chunksOf :: Int -> [a] -> a on chunksOf(k, xs)
script
on go(ys)
set {a, b} to splitAt(k, ys)
if isNull(a) then
{}
else
{a} & go(b)
end if
end go
end script
result's go(xs)
end chunksOf
-- compose :: (b -> c) -> (a -> b) -> (a -> c) on compose(f, g)
script
on |λ|(x)
mReturn(f)'s |λ|(mReturn(g)'s |λ|(x))
end |λ|
end script
end compose
-- concat :: a -> [a] | [String] -> String on concat(xs)
if length of xs > 0 and class of (item 1 of xs) is string then
set acc to ""
else
set acc to {}
end if
repeat with i from 1 to length of xs
set acc to acc & item i of xs
end repeat
acc
end concat
-- cons :: a -> [a] -> [a] on cons(x, xs)
{x} & xs
end cons
-- curry :: (Script|Handler) -> Script on curry(f)
script
on |λ|(a)
script
on |λ|(b)
|λ|(a, b) of mReturn(f)
end |λ|
end script
end |λ|
end script
end curry
-- enumFromTo :: Int -> Int -> [Int] on enumFromTo(m, n)
set lst to {}
repeat with i from m to n
set end of lst to i
end repeat
return lst
end enumFromTo
-- floor :: Num -> Int on floor(x)
if x < 0 and x mod 1 is not 0 then
(x div 1) - 1
else
(x div 1)
end if
end floor
-- foldl :: (a -> b -> a) -> a -> [b] -> a on foldl(f, startValue, xs)
tell mReturn(f)
set v to startValue
set lng to length of xs
repeat with i from 1 to lng
set v to |λ|(v, item i of xs, i, xs)
end repeat
return v
end tell
end foldl
-- foldr :: (b -> a -> a) -> a -> [b] -> a on foldr(f, startValue, xs)
tell mReturn(f)
set v to startValue
set lng to length of xs
repeat with i from lng to 1 by -1
set v to |λ|(item i of xs, v, i, xs)
end repeat
return v
end tell
end foldr
-- id :: a -> a on |id|(x)
x
end |id|
-- intercalate :: Text -> [Text] -> Text on intercalate(strText, lstText)
set {dlm, my text item delimiters} to {my text item delimiters, strText}
set strJoined to lstText as text
set my text item delimiters to dlm
return strJoined
end intercalate
-- isNull :: [a] -> Bool on isNull(xs)
if class of xs is string then
xs = ""
else
xs = {}
end if
end isNull
-- map :: (a -> b) -> [a] -> [b] on map(f, xs)
tell mReturn(f)
set lng to length of xs
set lst to {}
repeat with i from 1 to lng
set end of lst to |λ|(item i of xs, i, xs)
end repeat
return lst
end tell
end map
-- min :: Ord a => a -> a -> a on min(x, y)
if y < x then
y
else
x
end if
end min
-- Lift 2nd class handler function into 1st class script wrapper -- mReturn :: Handler -> Script on mReturn(f)
if class of f is script then
f
else
script
property |λ| : f
end script
end if
end mReturn
-- quot :: Int -> Int -> Int on quot(m, n)
m div n
end quot
-- range :: Ix a => (a, a) -> [a] on range({a, b})
if class of a is list then
set {xs, ys} to {a, b}
else
set {xs, ys} to {{a}, {b}}
end if
set lng to length of xs
if lng = length of ys then
if lng > 1 then
script
on |λ|(_, i)
enumFromTo(item i of xs, item i of ys)
end |λ|
end script
sequence(map(result, xs))
else
enumFromTo(a, b)
end if
else
{}
end if
end range
-- sequence :: Monad m => [m a] -> m [a] -- sequence :: [a] -> a on sequence(xs)
traverse(|id|, xs)
end sequence
-- show :: a -> String on show(e)
set c to class of e
if c = list then
script serialized
on |λ|(v)
show(v)
end |λ|
end script
"[" & intercalate(", ", map(serialized, e)) & "]"
else if c = record then
script showField
on |λ|(kv)
set {k, ev} to kv
"\"" & k & "\":" & show(ev)
end |λ|
end script
"{" & intercalate(", ", ¬
map(showField, zip(allKeys(e), allValues(e)))) & "}"
else if c = date then
"\"" & iso8601Z(e) & "\""
else if c = text then
"\"" & e & "\""
else if (c = integer or c = real) then
e as text
else if c = class then
"null"
else
try
e as text
on error
("«" & c as text) & "»"
end try
end if
end show
-- splitAt :: Int -> [a] -> ([a],[a]) on splitAt(n, xs)
if n > 0 and n < length of xs then
if class of xs is text then
{items 1 thru n of xs as text, items (n + 1) thru -1 of xs as text}
else
{items 1 thru n of xs, items (n + 1) thru -1 of xs}
end if
else
if n < 1 then
{{}, xs}
else
{xs, {}}
end if
end if
end splitAt
-- swap :: (a, b) -> (b, a) on swap(ab)
set {a, b} to ab
{b, a}
end swap
-- traverse :: (a -> [b]) -> [a] -> b on traverse(f, xs)
script
property mf : mReturn(f)
on |λ|(x, a)
|<*>|(map(curry(cons), mf's |λ|(x)), a)
end |λ|
end script
foldr(result, {{}}, xs)
end traverse
-- unlines :: [String] -> String on unlines(xs)
intercalate(linefeed, xs)
end unlines
-- zipWith :: (a -> b -> c) -> [a] -> [b] -> [c] on zipWith(f, xs, ys)
set lng to min(length of xs, length of ys)
set lst to {}
tell mReturn(f)
repeat with i from 1 to lng
set end of lst to |λ|(item i of xs, item i of ys)
end repeat
return lst
end tell
end zipWith</lang>
- Output:
Lower [1, 0, 0, 0, 0] [1, 1, 0, 0, 0] [1, 2, 1, 0, 0] [1, 3, 3, 1, 0] [1, 4, 6, 4, 1] Upper [1, 1, 1, 1, 1] [0, 1, 2, 3, 4] [0, 0, 1, 3, 6] [0, 0, 0, 1, 4] [0, 0, 0, 0, 1] Symmetric [1, 1, 1, 1, 1] [1, 2, 3, 4, 5] [1, 3, 6, 10, 15] [1, 4, 10, 20, 35] [1, 5, 15, 35, 70]
AutoHotkey
<lang AutoHotkey>n := 5 MsgBox, 262144, ,% "" . "Pascal upper-triangular :`n" show(Pascal_Upper(n)) . "`n`nPascal lower-triangular :`n" show(Pascal_Lower(n)) . "`n`nPascal symmetric:`n" show(Pascal_Symm(n)) return
show(obj){ for i, o in obj{ line := "" for j, v in o line .= v ", " res .= "[" Trim(line, ", ") "]`n," } return "[" Trim(res, "`n,") "]" }
Pascal_Upper(n){ obj := fillObj(n) loop % n obj[1, A_Index] := 1 loop % n-1 obj[A_Index+1, 1] := 0 for i, o in obj for j, v in o if !(i = 1 or j = 1) obj[i, j] := obj[i, j-1] + obj[i-1, j-1] return obj }
Pascal_Lower(n){ obj := fillObj(n) loop % n obj[A_Index, 1] := 1 loop % n-1 obj[1, A_Index+1] := 0 for i, o in obj for j, v in o if !(i = 1 or j = 1) obj[i, j] := obj[i-1, j] + obj[i-1, j-1] return obj }
Pascal_Symm(n){ obj := fillObj(n) loop % n obj[A_Index, 1] := 1 loop % n-1 obj[1, A_Index+1] := 1 for i, o in obj for j, v in o if !(i = 1 or j = 1) obj[i, j] := obj[i-1, j] + obj[i, j-1] return obj }
fillObj(n){ obj := [] loop % n{ i := A_Index loop % n obj[i, A_Index] := 0 } return obj }</lang>
- Output:
Pascal upper-triangular : [[1, 1, 1, 1, 1] ,[0, 1, 2, 3, 4] ,[0, 0, 1, 3, 6] ,[0, 0, 0, 1, 4] ,[0, 0, 0, 0, 1]] Pascal lower-triangular : [[1, 0, 0, 0, 0] ,[1, 1, 0, 0, 0] ,[1, 2, 1, 0, 0] ,[1, 3, 3, 1, 0] ,[1, 4, 6, 4, 1]] Pascal symmetric: [[1, 1, 1, 1, 1] ,[1, 2, 3, 4, 5] ,[1, 3, 6, 10, 15] ,[1, 4, 10, 20, 35] ,[1, 5, 15, 35, 70]]
BASIC
<lang basic>10 DEFINT A-Z: S=5: DIM M(S,S) 20 PRINT "Lower-triangular matrix:": GOSUB 200: GOSUB 100 30 PRINT "Upper-triangular matrix:": GOSUB 300: GOSUB 100 40 PRINT "Symmetric matrix:": GOSUB 400: GOSUB 100 50 END 100 REM *** Print the matrix M *** 110 FOR Y=1 TO S 120 FOR X=1 TO S 130 PRINT USING " ##";M(X,Y); 140 NEXT X 150 PRINT 160 NEXT Y 170 PRINT 180 RETURN 200 REM *** Generate the lower-triangular matrix *** 210 FOR X=1 TO S: FOR Y=1 TO S 220 ON -(X>Y)-2*(X=Y OR X=1) GOTO 240,250 230 M(X,Y)=M(X-1,Y-1)+M(X,Y-1): GOTO 260 240 M(X,Y)=0: GOTO 260 250 M(X,Y)=1: GOTO 260 260 NEXT Y,X 270 RETURN 300 REM *** Generate the upper-triangular matrix *** 310 FOR X=1 TO S: FOR Y=1 TO S 320 ON -(X<Y)-2*(X=Y OR Y=1) GOTO 340,350 330 M(X,Y)=M(X-1,Y-1)+M(X-1,Y): GOTO 360 340 M(X,Y)=0: GOTO 360 350 M(X,Y)=1: GOTO 360 360 NEXT Y,X 370 RETURN 400 REM *** Generate the symmetric matrix *** 410 FOR X=1 TO S: FOR Y=1 TO S 420 IF X=1 OR Y=1 THEN M(X,Y)=1 ELSE M(X,Y)=M(X-1,Y)+M(X,Y-1) 430 NEXT Y,X 440 RETURN</lang>
- Output:
Lower-triangular matrix: 1 0 0 0 0 1 1 0 0 0 1 2 1 0 0 1 3 3 1 0 1 4 6 4 1 Upper-triangular matrix: 1 1 1 1 1 0 1 2 3 4 0 0 1 3 6 0 0 0 1 4 0 0 0 0 1 Symmetric matrix: 1 1 1 1 1 1 2 3 4 5 1 3 6 10 15 1 4 10 20 35 1 5 15 35 70
BCPL
<lang bcpl>get "libhdr" manifest $( size = 5 $)
// Matrix index let ix(mat, n, x, y) = mat+y*n+x
let lower(m, n) be
for y=0 to n-1
for x=0 to n-1 do
!ix(m,n,x,y) :=
x>y -> 0,
x=y | x=0 -> 1,
!ix(m,n,x-1,y-1) + !ix(m,n,x,y-1)
let upper(m, n) be
for y=0 to n-1
for x=0 to n-1 do
!ix(m,n,x,y) :=
x<y -> 0,
x=y | y=0 -> 1,
!ix(m,n,x-1,y-1) + !ix(m,n,x-1,y)
let symmetric(m, n) be
for y=0 to n-1
for x=0 to n-1 do
!ix(m,n,x,y) :=
x=0 | y=0 -> 1,
!ix(m,n,x-1,y) + !ix(m,n,x,y-1)
// Print matrix let writemat(m, n, d) be
for y=0 to n-1
$( for x=0 to n-1
$( writed(!ix(m,n,x,y), d)
wrch(' ')
$)
wrch('*N')
$)
// Generate and print 5-by-5 matrices let start() be $( let mat = vec size * size
writes("Upper-triangular matrix:*N")
upper(mat, size) ; writemat(mat, size, 2)
writes("*NLower-triangular matrix:*N")
lower(mat, size) ; writemat(mat, size, 2)
writes("*NSymmetric matrix:*N")
symmetric(mat, size) ; writemat(mat, size, 2)
$)</lang>
- Output:
Upper-triangular matrix: 1 1 1 1 1 0 1 2 3 4 0 0 1 3 6 0 0 0 1 4 0 0 0 0 1 Lower-triangular matrix: 1 0 0 0 0 1 1 0 0 0 1 2 1 0 0 1 3 3 1 0 1 4 6 4 1 Symmetric matrix: 1 1 1 1 1 1 2 3 4 5 1 3 6 10 15 1 4 10 20 35 1 5 15 35 70
C
<lang c>
- include <stdio.h>
- include <stdlib.h>
void pascal_low(int **mat, int n) {
int i, j;
for (i = 0; i < n; ++i)
for (j = 0; j < n; ++j)
if (i < j)
mat[i][j] = 0;
else if (i == j || j == 0)
mat[i][j] = 1;
else
mat[i][j] = mat[i - 1][j - 1] + mat[i - 1][j];
}
void pascal_upp(int **mat, int n) {
int i, j;
for (i = 0; i < n; ++i)
for (j = 0; j < n; ++j)
if (i > j)
mat[i][j] = 0;
else if (i == j || i == 0)
mat[i][j] = 1;
else
mat[i][j] = mat[i - 1][j - 1] + mat[i][j - 1];
}
void pascal_sym(int **mat, int n) {
int i, j;
for (i = 0; i < n; ++i)
for (j = 0; j < n; ++j)
if (i == 0 || j == 0)
mat[i][j] = 1;
else
mat[i][j] = mat[i - 1][j] + mat[i][j - 1];
}
int main(int argc, char * argv[]) {
int **mat; int i, j, n;
/* Input size of the matrix */ n = 5;
/* Matrix allocation */
mat = calloc(n, sizeof(int *));
for (i = 0; i < n; ++i)
mat[i] = calloc(n, sizeof(int));
/* Matrix computation */
printf("=== Pascal upper matrix ===\n");
pascal_upp(mat, n);
for (i = 0; i < n; i++)
for (j = 0; j < n; j++)
printf("%4d%c", mat[i][j], j < n - 1 ? ' ' : '\n');
printf("=== Pascal lower matrix ===\n");
pascal_low(mat, n);
for (i = 0; i < n; i++)
for (j = 0; j < n; j++)
printf("%4d%c", mat[i][j], j < n - 1 ? ' ' : '\n');
printf("=== Pascal symmetric matrix ===\n");
pascal_sym(mat, n);
for (i = 0; i < n; i++)
for (j = 0; j < n; j++)
printf("%4d%c", mat[i][j], j < n - 1 ? ' ' : '\n');
return 0;
} </lang>
- Output:
=== Pascal upper matrix === 1 1 1 1 1 0 1 2 3 4 0 0 1 3 6 0 0 0 1 4 0 0 0 0 1 === Pascal lower matrix === 1 0 0 0 0 1 1 0 0 0 1 2 1 0 0 1 3 3 1 0 1 4 6 4 1 === Pascal symmetric matrix === 1 1 1 1 1 1 2 3 4 5 1 3 6 10 15 1 4 10 20 35 1 5 15 35 70
C#
<lang csharp>using System;
public static class PascalMatrixGeneration {
public static void Main() {
Print(GenerateUpper(5));
Console.WriteLine();
Print(GenerateLower(5));
Console.WriteLine();
Print(GenerateSymmetric(5));
}
static int[,] GenerateUpper(int size) {
int[,] m = new int[size, size];
for (int c = 0; c < size; c++) m[0, c] = 1;
for (int r = 1; r < size; r++) {
for (int c = r; c < size; c++) {
m[r, c] = m[r-1, c-1] + m[r, c-1];
}
}
return m;
}
static int[,] GenerateLower(int size) {
int[,] m = new int[size, size];
for (int r = 0; r < size; r++) m[r, 0] = 1;
for (int c = 1; c < size; c++) {
for (int r = c; r < size; r++) {
m[r, c] = m[r-1, c-1] + m[r-1, c];
}
}
return m;
}
static int[,] GenerateSymmetric(int size) {
int[,] m = new int[size, size];
for (int i = 0; i < size; i++) m[0, i] = m[i, 0] = 1;
for (int r = 1; r < size; r++) {
for (int c = 1; c < size; c++) {
m[r, c] = m[r-1, c] + m[r, c-1];
}
}
return m;
}
static void Print(int[,] matrix) {
string[,] m = ToString(matrix);
int width = m.Cast<string>().Select(s => s.Length).Max();
int rows = matrix.GetLength(0), columns = matrix.GetLength(1);
for (int row = 0; row < rows; row++) {
Console.WriteLine("|" + string.Join(" ", Range(0, columns).Select(column => m[row, column].PadLeft(width, ' '))) + "|");
}
}
static string[,] ToString(int[,] matrix) {
int rows = matrix.GetLength(0), columns = matrix.GetLength(1);
string[,] m = new string[rows, columns];
for (int r = 0; r < rows; r++) {
for (int c = 0; c < columns; c++) {
m[r, c] = matrix[r, c].ToString();
}
}
return m;
}
}</lang>
- Output:
|1 1 1 1 1| |0 1 2 3 4| |0 0 1 3 6| |0 0 0 1 4| |0 0 0 0 1| |1 0 0 0 0| |1 1 0 0 0| |1 2 1 0 0| |1 3 3 1 0| |1 4 6 4 1| | 1 1 1 1 1| | 1 2 3 4 5| | 1 3 6 10 15| | 1 4 10 20 35| | 1 5 15 35 70|
C++
<lang cpp>#include <iostream>
- include <vector>
typedef std::vector<std::vector<int>> vv;
vv pascal_upper(int n) {
vv matrix(n);
for (int i = 0; i < n; ++i) {
for (int j = 0; j < n; ++j) {
if (i > j) matrix[i].push_back(0);
else if (i == j || i == 0) matrix[i].push_back(1);
else matrix[i].push_back(matrix[i - 1][j - 1] + matrix[i][j - 1]);
}
}
return matrix;
}
vv pascal_lower(int n) {
vv matrix(n);
for (int i = 0; i < n; ++i) {
for (int j = 0; j < n; ++j) {
if (i < j) matrix[i].push_back(0);
else if (i == j || j == 0) matrix[i].push_back(1);
else matrix[i].push_back(matrix[i - 1][j - 1] + matrix[i - 1][j]);
}
}
return matrix;
}
vv pascal_symmetric(int n) {
vv matrix(n);
for (int i = 0; i < n; ++i) {
for (int j = 0; j < n; ++j) {
if (i == 0 || j == 0) matrix[i].push_back(1);
else matrix[i].push_back(matrix[i][j - 1] + matrix[i - 1][j]);
}
}
return matrix;
}
void print_matrix(vv matrix) {
for (std::vector<int> v: matrix) {
for (int i: v) {
std::cout << " " << i;
}
std::cout << std::endl;
}
}
int main() {
std::cout << "PASCAL UPPER MATRIX" << std::endl; print_matrix(pascal_upper(5)); std::cout << "PASCAL LOWER MATRIX" << std::endl; print_matrix(pascal_lower(5)); std::cout << "PASCAL SYMMETRIC MATRIX" << std::endl; print_matrix(pascal_symmetric(5));
}</lang>
- Output:
PASCAL UPPER MATRIX 1 1 1 1 1 0 1 2 3 4 0 0 1 3 6 0 0 0 1 4 0 0 0 0 1 PASCAL LOWER MATRIX 1 0 0 0 0 1 1 0 0 0 1 2 1 0 0 1 3 3 1 0 1 4 6 4 1 PASCAL SYMMETRIC MATRIX 1 1 1 1 1 1 2 3 4 5 1 3 6 10 15 1 4 10 20 35 1 5 15 35 70
Clojure
<lang clojure>(defn binomial-coeff [n k]
(reduce #(quot (* %1 (inc (- n %2))) %2)
1
(range 1 (inc k))))
(defn pascal-upper [n]
(map
(fn [i]
(map (fn [j]
(binomial-coeff j i))
(range n)))
(range n)))
(defn pascal-lower [n]
(map
(fn [i]
(map (fn [j]
(binomial-coeff i j))
(range n)))
(range n)))
(defn pascal-symmetric [n]
(map
(fn [i]
(map (fn [j]
(binomial-coeff (+ i j) i))
(range n)))
(range n)))
(defn pascal-matrix [n]
(println "Upper:") (run! println (pascal-upper n)) (println) (println "Lower:") (run! println (pascal-lower n)) (println) (println "Symmetric:") (run! println (pascal-symmetric n)))</lang>
- Output:
=> (pascal-matrix 5) Upper: (1 1 1 1 1) (0 1 2 3 4) (0 0 1 3 6) (0 0 0 1 4) (0 0 0 0 1) Lower: (1 0 0 0 0) (1 1 0 0 0) (1 2 1 0 0) (1 3 3 1 0) (1 4 6 4 1) Symmetric: (1 1 1 1 1) (1 2 3 4 5) (1 3 6 10 15) (1 4 10 20 35) (1 5 15 35 70)
CLU
<lang clu>matrix = array[array[int]]
make_matrix = proc (gen: proctype (int,int,matrix) returns (int),
size: int) returns (matrix)
m: matrix := matrix$fill_copy(0, size, array[int]$fill(0, size, 0))
for y: int in int$from_to(0, size-1) do
for x: int in int$from_to(0, size-1) do
m[y][x] := gen(x,y,m)
end
end
return(m)
end make_matrix
lower = proc (x,y: int, m: matrix) returns (int)
if x>y then return(0) elseif x=y | x=0 then return(1) else return( m[y-1][x-1] + m[y-1][x] ) end
end lower
upper = proc (x,y: int, m: matrix) returns (int)
if x<y then return(0) elseif x=y | y=0 then return(1) else return( m[y-1][x-1] + m[y][x-1] ) end
end upper
symmetric = proc (x,y: int, m: matrix) returns (int)
if x=0 | y=0 then return(1) else return(m[y][x-1] + m[y-1][x]) end
end symmetric
print_matrix = proc (s: stream, m: matrix, w: int)
for line: array[int] in matrix$elements(m) do
for item: int in array[int]$elements(line) do
stream$putright(s, int$unparse(item), w)
stream$putc(s, ' ')
end
stream$putl(s, "")
end
end print_matrix
start_up = proc ()
po: stream := stream$primary_output() stream$putl(po, "Upper-triangular matrix:") print_matrix(po, make_matrix(upper,5), 1) stream$putl(po, "\nLower-triangular matrix:") print_matrix(po, make_matrix(lower,5), 1) stream$putl(po, "\nSymmetric matrix:") print_matrix(po, make_matrix(symmetric,5), 2)
end start_up</lang>
- Output:
Upper-triangular matrix: 1 1 1 1 1 0 1 2 3 4 0 0 1 3 6 0 0 0 1 4 0 0 0 0 1 Lower-triangular matrix: 1 0 0 0 0 1 1 0 0 0 1 2 1 0 0 1 3 3 1 0 1 4 6 4 1 Symmetric matrix: 1 1 1 1 1 1 2 3 4 5 1 3 6 10 15 1 4 10 20 35 1 5 15 35 70
Common Lisp
<lang lisp>(defun pascal-lower (n &aux (a (make-array (list n n) :initial-element 0)))
(dotimes (i n)
(setf (aref a i 0) 1))
(dotimes (i (1- n) a)
(dotimes (j (1- n))
(setf (aref a (1+ i) (1+ j))
(+ (aref a i j)
(aref a i (1+ j)))))))
(defun pascal-upper (n &aux (a (make-array (list n n) :initial-element 0)))
(dotimes (i n)
(setf (aref a 0 i) 1))
(dotimes (i (1- n) a)
(dotimes (j (1- n))
(setf (aref a (1+ j) (1+ i))
(+ (aref a j i)
(aref a (1+ j) i))))))
(defun pascal-symmetric (n &aux (a (make-array (list n n) :initial-element 0)))
(dotimes (i n)
(setf (aref a i 0) 1 (aref a 0 i) 1))
(dotimes (i (1- n) a)
(dotimes (j (1- n))
(setf (aref a (1+ i) (1+ j))
(+ (aref a (1+ i) j)
(aref a i (1+ j)))))))
? (pascal-lower 4)
- 2A((1 0 0 0) (1 1 0 0) (1 2 1 0) (1 3 3 1))
? (pascal-upper 4)
- 2A((1 1 1 1) (0 1 2 3) (0 0 1 3) (0 0 0 1))
? (pascal-symmetric 4)
- 2A((1 1 1 1) (1 2 3 4) (1 3 6 10) (1 4 10 20))
- In case one really insists in printing the array row by row
(defun print-matrix (a)
(let ((p (array-dimension a 0))
(q (array-dimension a 1)))
(dotimes (i p)
(dotimes (j q)
(princ (aref a i j))
(princ #\Space))
(terpri))))
? (print-matrix (pascal-lower 5)) 1 0 0 0 0 1 1 0 0 0 1 2 1 0 0 1 3 3 1 0 1 4 6 4 1
? (print-matrix (pascal-upper 5)) 1 1 1 1 1 0 1 2 3 4 0 0 1 3 6 0 0 0 1 4 0 0 0 0 1
? (print-matrix (pascal-symmetric 5)) 1 1 1 1 1 1 2 3 4 5 1 3 6 10 15 1 4 10 20 35 1 5 15 35 70</lang>
D
<lang d>import std.stdio, std.bigint, std.range, std.algorithm;
auto binomialCoeff(in uint n, in uint k) pure nothrow {
BigInt result = 1;
foreach (immutable i; 1 .. k + 1)
result = result * (n - i + 1) / i;
return result;
}
auto pascalUpp(in uint n) pure nothrow {
return n.iota.map!(i => n.iota.map!(j => binomialCoeff(j, i)));
}
auto pascalLow(in uint n) pure nothrow {
return n.iota.map!(i => n.iota.map!(j => binomialCoeff(i, j)));
}
auto pascalSym(in uint n) pure nothrow {
return n.iota.map!(i => n.iota.map!(j => binomialCoeff(i + j, i)));
}
void main() {
enum n = 5;
writefln("Upper:\n%(%(%2d %)\n%)", pascalUpp(n));
writefln("\nLower:\n%(%(%2d %)\n%)", pascalLow(n));
writefln("\nSymmetric:\n%(%(%2d %)\n%)", pascalSym(n));
}</lang>
- Output:
Upper: 1 1 1 1 1 0 1 2 3 4 0 0 1 3 6 0 0 0 1 4 0 0 0 0 1 Lower: 1 0 0 0 0 1 1 0 0 0 1 2 1 0 0 1 3 3 1 0 1 4 6 4 1 Symmetric: 1 1 1 1 1 1 2 3 4 5 1 3 6 10 15 1 4 10 20 35 1 5 15 35 70
Delphi
See Pascal.
Elixir
<lang elixir>defmodule Pascal do
defp ij(n), do: for i <- 1..n, j <- 1..n, do: {i,j}
def upper_triangle(n) do
Enum.reduce(ij(n), Map.new, fn {i,j},acc ->
val = cond do
i==1 -> 1
j 0
true -> Map.get(acc, {i-1, j-1}) + Map.get(acc, {i, j-1})
end
Map.put(acc, {i,j}, val)
end) |> print(1..n)
end
def lower_triangle(n) do
Enum.reduce(ij(n), Map.new, fn {i,j},acc ->
val = cond do
j==1 -> 1
i<j -> 0
true -> Map.get(acc, {i-1, j-1}) + Map.get(acc, {i-1, j})
end
Map.put(acc, {i,j}, val)
end) |> print(1..n)
end
def symmetic_triangle(n) do
Enum.reduce(ij(n), Map.new, fn {i,j},acc ->
val = if i==1 or j==1, do: 1,
else: Map.get(acc, {i-1, j}) + Map.get(acc, {i, j-1})
Map.put(acc, {i,j}, val)
end) |> print(1..n)
end
def print(matrix, range) do
Enum.each(range, fn i ->
Enum.map(range, fn j -> Map.get(matrix, {i,j}) end) |> IO.inspect
end)
end
end
IO.puts "Pascal upper-triangular matrix:" Pascal.upper_triangle(5) IO.puts "Pascal lower-triangular matrix:" Pascal.lower_triangle(5) IO.puts "Pascal symmetric matrix:" Pascal.symmetic_triangle(5)</lang>
- Output:
Pascal upper-triangular matrix: [1, 1, 1, 1, 1] [0, 1, 2, 3, 4] [0, 0, 1, 3, 6] [0, 0, 0, 1, 4] [0, 0, 0, 0, 1] Pascal lower-triangular matrix: [1, 0, 0, 0, 0] [1, 1, 0, 0, 0] [1, 2, 1, 0, 0] [1, 3, 3, 1, 0] [1, 4, 6, 4, 1] Pascal symmetric matrix: [1, 1, 1, 1, 1] [1, 2, 3, 4, 5] [1, 3, 6, 10, 15] [1, 4, 10, 20, 35] [1, 5, 15, 35, 70]
Excel
LAMBDA
Binding the names PASCALMATRIX, BINCOEFF and SYMMETRIC to the following lambda expressions in the Name Manager of the Excel WorkBook:
(See LAMBDA: The ultimate Excel worksheet function)
<lang lisp>PASCALMATRIX =LAMBDA(n,
LAMBDA(f,
LET(
ixs, SEQUENCE(n, n, 0, 1),
f(BINCOEFF)(
QUOTIENT(ixs, n)
)(
MOD(ixs, n)
)
)
)
)
BINCOEFF
=LAMBDA(n,
LAMBDA(k,
IF(n < k,
0,
QUOTIENT(FACT(n), FACT(k) * FACT(n - k))
)
)
)
SYMMETRIC
=LAMBDA(f,
LAMBDA(a,
LAMBDA(b,
f(a + b)(b)
)
)
)</lang>
and also assuming the following generic bindings in the Name Manager for the WorkBook:
<lang lisp>FLIP =LAMBDA(f,
LAMBDA(a,
LAMBDA(b,
f(b)(a)
)
)
)
ID
=LAMBDA(x, x)</lang>
- Output:
| fx | =PASCALMATRIX(5)( ID ) | ||||||||
|---|---|---|---|---|---|---|---|---|---|
| A | B | C | D | E | F | G | H | ||
| 1 | Lower | ||||||||
| 2 | 1 | 0 | 0 | 0 | 0 | ||||
| 3 | 1 | 1 | 0 | 0 | 0 | ||||
| 4 | 1 | 2 | 1 | 0 | 0 | ||||
| 5 | 1 | 3 | 3 | 1 | 0 | ||||
| 6 | 1 | 4 | 6 | 4 | 1 | ||||
| fx | =PASCALMATRIX(5)( FLIP ) | ||||||||
|---|---|---|---|---|---|---|---|---|---|
| A | B | C | D | E | F | G | H | ||
| 1 | Upper | ||||||||
| 2 | 1 | 1 | 1 | 1 | 1 | ||||
| 3 | 0 | 1 | 2 | 3 | 4 | ||||
| 4 | 0 | 0 | 1 | 3 | 6 | ||||
| 5 | 0 | 0 | 0 | 1 | 4 | ||||
| 6 | 0 | 0 | 0 | 0 | 1 | ||||
| fx | =PASCALMATRIX(5)( SYMMETRIC ) | ||||||||
|---|---|---|---|---|---|---|---|---|---|
| A | B | C | D | E | F | G | H | ||
| 1 | Symmetric | ||||||||
| 2 | 1 | 1 | 1 | 1 | 1 | ||||
| 3 | 1 | 2 | 3 | 4 | 5 | ||||
| 4 | 1 | 3 | 6 | 10 | 15 | ||||
| 5 | 1 | 4 | 10 | 20 | 35 | ||||
| 6 | 1 | 5 | 15 | 35 | 70 | ||||
Factor
<lang factor>USING: arrays fry io kernel math math.combinatorics math.matrices prettyprint sequences ;
- pascal ( n quot -- m )
[ dup 2array <coordinate-matrix> ] dip '[ first2 @ nCk ] matrix-map ; inline
- lower ( n -- m ) [ ] pascal ;
- upper ( n -- m ) lower flip ;
- symmetric ( n -- m ) [ [ + ] keep ] pascal ;
5 [ lower "Lower:" ] [ upper "Upper:" ] [ symmetric "Symmetric:" ] tri [ print simple-table. nl ] 2tri@</lang>
- Output:
Lower: 1 0 0 0 0 1 1 0 0 0 1 2 1 0 0 1 3 3 1 0 1 4 6 4 1 Upper: 1 1 1 1 1 0 1 2 3 4 0 0 1 3 6 0 0 0 1 4 0 0 0 0 1 Symmetric: 1 1 1 1 1 1 2 3 4 5 1 3 6 10 15 1 4 10 20 35 1 5 15 35 70
Fermat
<lang fermat>&a; {set mode to 0-indexed matrices} Func Pasmat( n, t ) =
;{create a Pascal matrix of size n by n}
;{t=0 -> upper triangular, 1 -> lower triangular,2->symmetric}
Array m[n, n]; {result is stored in array m}
if t = 0 then
[m]:=[<i=0,n-1><j=0,n-1> Bin(j,i) ];
fi;
if t = 1 then
[m]:=[<i=0,n-1><j=0,n-1> Bin(i,j) ];
fi;
if t = 2 then
[m]:=[<i=0,n-1><j=0,n-1> Bin(i+j,i) ];
fi;
.;
Pasmat(5, 0); !!([m); !; Pasmat(5, 1); !!([m); !; Pasmat(5, 2); !!([m);</lang>
- Output:
1, 1, 1, 1, 1, 0, 1, 2, 3, 4, 0, 0, 1, 3, 6, 0, 0, 0, 1, 4, 0, 0, 0, 0, 1
1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 2, 1, 0, 0, 1, 3, 3, 1, 0, 1, 4, 6, 4, 1
1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 1, 3, 6, 10, 15, 1, 4, 10, 20, 35, 1, 5, 15, 35, 70
Fortran
The following program uses features of Fortran 2003.
<lang fortran>module pascal
implicit none
contains
function pascal_lower(n) result(a)
integer :: n, i, j
integer, allocatable :: a(:, :)
allocate(a(n, n))
a = 0
do i = 1, n
a(i, 1) = 1
end do
do i = 2, n
do j = 2, i
a(i, j) = a(i - 1, j) + a(i - 1, j - 1)
end do
end do
end function
function pascal_upper(n) result(a)
integer :: n, i, j
integer, allocatable :: a(:, :)
allocate(a(n, n))
a = 0
do i = 1, n
a(1, i) = 1
end do
do i = 2, n
do j = 2, i
a(j, i) = a(j, i - 1) + a(j - 1, i - 1)
end do
end do
end function
function pascal_symmetric(n) result(a)
integer :: n, i, j
integer, allocatable :: a(:, :)
allocate(a(n, n))
a = 0
do i = 1, n
a(i, 1) = 1
a(1, i) = 1
end do
do i = 2, n
do j = 2, n
a(i, j) = a(i - 1, j) + a(i, j - 1)
end do
end do
end function
subroutine print_matrix(a)
integer :: a(:, :)
integer :: n, i
n = ubound(a, 1)
do i = 1, n
print *, a(i, :)
end do
end subroutine
end module
program ex_pascal
use pascal implicit none integer :: n integer, allocatable :: a(:, :) print *, "Size?" read *, n print *, "Lower Pascal Matrix" a = pascal_lower(n) call print_matrix(a) print *, "Upper Pascal Matrix" a = pascal_upper(n) call print_matrix(a) print *, "Symmetric Pascal Matrix" a = pascal_symmetric(n) call print_matrix(a)
end program</lang>
<lang> Size? 5
Lower Pascal Matrix
1 0 0 0 0
1 1 0 0 0
1 2 1 0 0
1 3 3 1 0
1 4 6 4 1
Upper Pascal Matrix
1 1 1 1 1
0 1 2 3 4
0 0 1 3 6
0 0 0 1 4
0 0 0 0 1
Symmetric Pascal Matrix
1 1 1 1 1
1 2 3 4 5
1 3 6 10 15
1 4 10 20 35
1 5 15 35 70</lang>
FreeBASIC
<lang freebasic> sub print_matrix( M() as integer )
'displays a matrix
for row as integer = 0 to ubound(M, 1)
for col as integer = 0 to ubound(M, 2)
print using "#### ";M(row, col);
next col
print
next row
return
end sub
function fact( n as uinteger ) as uinteger
'quick and dirty factorial if n<2 then return 1 else return n*fact(n-1)
end function
function nCp( n as uinteger, p as uinteger ) as uinteger
'quick and dirty binomial if p>n then return 0 else return fact(n)/(fact(p)*fact(n-p))
end function
sub make_pascal( M() as integer, typ as const ubyte )
'allocate the matrix first
'typ 0 = jCi, 1=iCj, 2=(j+i)Ci
for i as uinteger = 0 to ubound(M,1)
for j as uinteger = 0 to ubound(M,2)
select case typ
case 0
M(i,j) = nCp(j, i)
case 1
M(i,j) = nCp(i, j)
case 2
M(i,j) = nCp(i + j, j)
case else
M(i, j) = 0
end select
next j
next i
return
end sub
dim as integer M(0 to 4, 0 to 4) print "Upper triangular" make_pascal( M(), 0 ) print_matrix( M() ) print "Lower triangular" make_pascal( M(), 1 ) print_matrix( M() ) print "Symmetric" make_pascal( M(), 2 ) print_matrix( M() ) print "Technically the matrix needn't be square :)" dim as integer Q(0 to 4, 0 to 9) make_pascal( Q(), 2 ) print_matrix( Q() )</lang>
- Output:
Upper triangular
1 1 1 1 1 0 1 2 3 4 0 0 1 3 6 0 0 0 1 4 0 0 0 0 1Lower triangular
1 0 0 0 0 1 1 0 0 0 1 2 1 0 0 1 3 3 1 0 1 4 6 4 1Symmetric
1 1 1 1 1 1 2 3 4 5 1 3 6 10 15 1 4 10 20 35 1 5 15 35 70Technically the matrix needn't be square :)
1 1 1 1 1 1 1 1 1 1 1 2 3 4 5 6 7 8 9 10 1 3 6 10 15 21 28 36 45 55 1 4 10 20 35 56 84 120 165 220 1 5 15 35 70 126 210 330 495 715
Go
<lang go>package main
import (
"fmt" "strings"
)
func binomial(n, k int) int {
if n < k {
return 0
}
if n == 0 || k == 0 {
return 1
}
num := 1
for i := k + 1; i <= n; i++ {
num *= i
}
den := 1
for i := 2; i <= n-k; i++ {
den *= i
}
return num / den
}
func pascalUpperTriangular(n int) [][]int {
m := make([][]int, n)
for i := 0; i < n; i++ {
m[i] = make([]int, n)
for j := 0; j < n; j++ {
m[i][j] = binomial(j, i)
}
}
return m
}
func pascalLowerTriangular(n int) [][]int {
m := make([][]int, n)
for i := 0; i < n; i++ {
m[i] = make([]int, n)
for j := 0; j < n; j++ {
m[i][j] = binomial(i, j)
}
}
return m
}
func pascalSymmetric(n int) [][]int {
m := make([][]int, n)
for i := 0; i < n; i++ {
m[i] = make([]int, n)
for j := 0; j < n; j++ {
m[i][j] = binomial(i+j, i)
}
}
return m
}
func printMatrix(title string, m [][]int) {
n := len(m)
fmt.Println(title)
fmt.Print("[")
for i := 0; i < n; i++ {
if i > 0 {
fmt.Print(" ")
}
mi := strings.Replace(fmt.Sprint(m[i]), " ", ", ", -1)
fmt.Print(mi)
if i < n-1 {
fmt.Println(",")
} else {
fmt.Println("]\n")
}
}
}
func main() {
printMatrix("Pascal upper-triangular matrix", pascalUpperTriangular(5))
printMatrix("Pascal lower-triangular matrix", pascalLowerTriangular(5))
printMatrix("Pascal symmetric matrix", pascalSymmetric(5))
}</lang>
- Output:
Pascal upper-triangular matrix [[1, 1, 1, 1, 1], [0, 1, 2, 3, 4], [0, 0, 1, 3, 6], [0, 0, 0, 1, 4], [0, 0, 0, 0, 1]] Pascal lower-triangular matrix [[1, 0, 0, 0, 0], [1, 1, 0, 0, 0], [1, 2, 1, 0, 0], [1, 3, 3, 1, 0], [1, 4, 6, 4, 1]] Pascal symmetric matrix [[1, 1, 1, 1, 1], [1, 2, 3, 4, 5], [1, 3, 6, 10, 15], [1, 4, 10, 20, 35], [1, 5, 15, 35, 70]]
Haskell
<lang haskell>import Data.List (transpose) import System.Environment (getArgs) import Text.Printf (printf)
-- Pascal's triangle. pascal :: Int pascal = iterate (\row -> 1 : zipWith (+) row (tail row) ++ [1]) [1]
-- The n by n Pascal lower triangular matrix. pascLow :: Int -> Int pascLow n = zipWith (\row i -> row ++ replicate (n-i) 0) (take n pascal) [1..]
-- The n by n Pascal upper triangular matrix. pascUp :: Int -> Int pascUp = transpose . pascLow
-- The n by n Pascal symmetric matrix. pascSym :: Int -> Int pascSym n = take n . map (take n) . transpose $ pascal
-- Format and print a matrix. printMat :: String -> Int -> IO () printMat title mat = do
putStrLn $ title ++ "\n" mapM_ (putStrLn . concatMap (printf " %2d")) mat putStrLn "\n"
main :: IO () main = do
ns <- fmap (map read) getArgs
case ns of
[n] -> do printMat "Lower triangular" $ pascLow n
printMat "Upper triangular" $ pascUp n
printMat "Symmetric" $ pascSym n
_ -> error "Usage: pascmat <number>"</lang>
- Output:
Lower triangular 1 0 0 0 0 1 1 0 0 0 1 2 1 0 0 1 3 3 1 0 1 4 6 4 1 Upper triangular 1 1 1 1 1 0 1 2 3 4 0 0 1 3 6 0 0 0 1 4 0 0 0 0 1 Symmetric 1 1 1 1 1 1 2 3 4 5 1 3 6 10 15 1 4 10 20 35 1 5 15 35 70
Or, in terms of binomial coefficients and coordinate transformations:
<lang haskell>import Control.Monad (join) import Data.Bifunctor (bimap) import Data.Ix (range) import Data.List.Split (chunksOf) import Data.Tuple (swap)
PASCAL MATRIX ---------------------
pascalMatrix :: ((Int, Int) -> (Int, Int)) -> Int -> [Int] pascalMatrix f n =
bc . f
<$> range
((0, 0), join bimap pred (n, n))
-- Binomial coefficient bc :: (Int, Int) -> Int bc (n, k) =
foldr (\x a -> quot (a * succ (n - x)) x) 1 [k, pred k .. 1]
TEST -------------------------
matrixSize = 5 :: Int
main :: IO () main =
mapM_
putStrLn
( unlines
. ( \(s, xs) ->
s :
(show <$> chunksOf matrixSize xs)
)
<$> zip
["Lower", "Upper", "Symmetric"]
( pascalMatrix
<$> [ id, -- Lower
swap, -- Upper
\(a, b) -> (a + b, b) -- Symmetric
]
<*> [matrixSize]
)
)</lang>
- Output:
Lower [1,0,0,0,0] [1,1,0,0,0] [1,2,1,0,0] [1,3,3,1,0] [1,4,6,4,1] Upper [1,1,1,1,1] [0,1,2,3,4] [0,0,1,3,6] [0,0,0,1,4] [0,0,0,0,1] Symmetric [1,1,1,1,1] [1,2,3,4,5] [1,3,6,10,15] [1,4,10,20,35] [1,5,15,35,70]
J
<lang J> !/~ i. 5 1 1 1 1 1 0 1 2 3 4 0 0 1 3 6 0 0 0 1 4 0 0 0 0 1
!~/~ i. 5
1 0 0 0 0 1 1 0 0 0 1 2 1 0 0 1 3 3 1 0 1 4 6 4 1
(["0/ ! +/)~ i. 5
1 1 1 1 1 1 2 3 4 5 1 3 6 10 15 1 4 10 20 35 1 5 15 35 70</lang>
Explanation:
x!y is the number of ways of picking x balls (unordered) from a bag of y balls and x!/y for list x and list y gives a table where rows correspond to the elements of x and the columns correspond to the elements of y. Meanwhile !/~y is equivalent to y!/y (and i.y just counts the first y non-negative integers).
Also, x!~y is y!x (and the second example otherwise follows the same pattern as the first example.
For the final example we use an unadorned ! but prepare tables for its x and y values. Its right argument is a sum table, and its left argument is a left identity table. They look like this:
<lang J> (+/)~ i. 5 0 1 2 3 4 1 2 3 4 5 2 3 4 5 6 3 4 5 6 7 4 5 6 7 8
(["0/)~ i. 5
0 0 0 0 0 1 1 1 1 1 2 2 2 2 2 3 3 3 3 3 4 4 4 4 4</lang>
The parenthesis in these last two examples are redundant - they could have been omitted without changing the result, but were left in place for emphasis.
Java
<lang java>import static java.lang.System.out; import java.util.List; import java.util.function.Function; import java.util.stream.*; import static java.util.stream.Collectors.toList; import static java.util.stream.IntStream.range;
public class PascalMatrix {
static int binomialCoef(int n, int k) {
int result = 1;
for (int i = 1; i <= k; i++)
result = result * (n - i + 1) / i;
return result;
}
static List<IntStream> pascal(int n, Function<Integer, IntStream> f) {
return range(0, n).mapToObj(i -> f.apply(i)).collect(toList());
}
static List<IntStream> pascalUpp(int n) {
return pascal(n, i -> range(0, n).map(j -> binomialCoef(j, i)));
}
static List<IntStream> pascalLow(int n) {
return pascal(n, i -> range(0, n).map(j -> binomialCoef(i, j)));
}
static List<IntStream> pascalSym(int n) {
return pascal(n, i -> range(0, n).map(j -> binomialCoef(i + j, i)));
}
static void print(String label, List<IntStream> result) {
out.println("\n" + label);
for (IntStream row : result) {
row.forEach(i -> out.printf("%2d ", i));
System.out.println();
}
}
public static void main(String[] a) {
print("Upper: ", pascalUpp(5));
print("Lower: ", pascalLow(5));
print("Symmetric:", pascalSym(5));
}
}</lang>
Upper: 1 1 1 1 1 0 1 2 3 4 0 0 1 3 6 0 0 0 1 4 0 0 0 0 1 Lower: 1 0 0 0 0 1 1 0 0 0 1 2 1 0 0 1 3 3 1 0 1 4 6 4 1 Symmetric: 1 1 1 1 1 1 2 3 4 5 1 3 6 10 15 1 4 10 20 35 1 5 15 35 70
JavaScript
In terms of a binomial coefficient, and a function on a coordinate pair.
ES6
<lang JavaScript>(() => {
'use strict';
// -------------------PASCAL MATRIX--------------------
// pascalMatrix :: ((Int, Int) -> (Int, Int)) ->
// Int -> [Int]
const pascalMatrix = f =>
n => map(compose(binomialCoefficient, f))(
range([0, 0], [n - 1, n - 1])
);
// binomialCoefficient :: (Int, Int) -> Int
const binomialCoefficient = nk => {
const [n, k] = Array.from(nk);
return enumFromThenTo(k)(
pred(k)
)(1).reduceRight((a, x) => quot(
a * succ(n - x)
)(x), 1);
};
// ------------------------TEST------------------------
// main :: IO ()
const main = () => {
const matrixSize = 5;
console.log(intercalate('\n\n')(
zipWith(
k => xs => k + ':\n' + showMatrix(matrixSize)(xs)
)(['Lower', 'Upper', 'Symmetric'])(
apList(
map(pascalMatrix)([
identity, // Lower
swap, // Upper
([a, b]) => [a + b, b] // Symmetric
])
)([matrixSize])
)
));
};
// ----------------------DISPLAY-----------------------
// showMatrix :: Int -> [Int] -> String
const showMatrix = n =>
xs => {
const
ks = map(str)(xs),
w = maximum(map(length)(ks));
return unlines(
map(unwords)(chunksOf(n)(
map(justifyRight(w)(' '))(ks)
))
);
};
// -----------------GENERIC FUNCTIONS------------------
// Tuple (,) :: a -> b -> (a, b)
const Tuple = a =>
b => ({
type: 'Tuple',
'0': a,
'1': b,
length: 2
});
// apList (<*>) :: [(a -> b)] -> [a] -> [b]
const apList = fs =>
// The sequential application of each of a list
// of functions to each of a list of values.
xs => fs.flatMap(
f => xs.map(f)
);
// chunksOf :: Int -> [a] -> a const chunksOf = n => xs => enumFromThenTo(0)(n)( xs.length - 1 ).reduce( (a, i) => a.concat([xs.slice(i, (n + i))]), [] );
// compose (<<<) :: (b -> c) -> (a -> b) -> a -> c
const compose = (...fs) =>
x => fs.reduceRight((a, f) => f(a), x);
// concat :: a -> [a] // concat :: [String] -> String const concat = xs => 0 < xs.length ? ( xs.every(x => 'string' === typeof x) ? ( ) : [] ).concat(...xs) : xs;
// cons :: a -> [a] -> [a]
const cons = x =>
xs => [x].concat(xs);
// enumFromThenTo :: Int -> Int -> Int -> [Int]
const enumFromThenTo = x1 =>
x2 => y => {
const d = x2 - x1;
return Array.from({
length: Math.floor(y - x2) / d + 2
}, (_, i) => x1 + (d * i));
};
// enumFromTo :: Int -> Int -> [Int]
const enumFromTo = m =>
n => Array.from({
length: 1 + n - m
}, (_, i) => m + i);
// fst :: (a, b) -> a
const fst = tpl =>
// First member of a pair.
tpl[0];
// identity :: a -> a
const identity = x =>
// The identity function. (`id`, in Haskell)
x;
// intercalate :: String -> [String] -> String
const intercalate = s =>
// The concatenation of xs
// interspersed with copies of s.
xs => xs.join(s);
// justifyRight :: Int -> Char -> String -> String
const justifyRight = n =>
// The string s, preceded by enough padding (with
// the character c) to reach the string length n.
c => s => n > s.length ? (
s.padStart(n, c)
) : s;
// length :: [a] -> Int
const length = xs =>
// Returns Infinity over objects without finite
// length. This enables zip and zipWith to choose
// the shorter argument when one is non-finite,
// like cycle, repeat etc
(Array.isArray(xs) || 'string' === typeof xs) ? (
xs.length
) : Infinity;
// liftA2List :: (a -> b -> c) -> [a] -> [b] -> [c]
const liftA2List = f => xs => ys =>
// The binary operator f lifted to a function over two
// lists. f applied to each pair of arguments in the
// cartesian product of xs and ys.
xs.flatMap(
x => ys.map(f(x))
);
// map :: (a -> b) -> [a] -> [b]
const map = f =>
// The list obtained by applying f to each element of xs.
// (The image of xs under f).
xs => (Array.isArray(xs) ? (
xs
) : xs.split()).map(f);
// maximum :: Ord a => [a] -> a
const maximum = xs =>
// The largest value in a non-empty list.
0 < xs.length ? (
xs.slice(1).reduce(
(a, x) => x > a ? (
x
) : a, xs[0]
)
) : undefined;
// pred :: Enum a => a -> a
const pred = x =>
x - 1;
// quot :: Int -> Int -> Int const quot = n => m => Math.floor(n / m);
// The list of values in the subrange defined by a bounding pair.
// range([0, 2]) -> [0,1,2] // range([[0,0], [2,2]]) // -> [[0,0],[0,1],[0,2],[1,0],[1,1],[1,2],[2,0],[2,1],[2,2]] // range([[0,0,0],[1,1,1]]) // -> [[0,0,0],[0,0,1],[0,1,0],[0,1,1],[1,0,0],[1,0,1],[1,1,0],[1,1,1]]
// range :: Ix a => (a, a) -> [a]
function range() {
const
args = Array.from(arguments),
ab = 1 !== args.length ? (
args
) : args[0],
[as, bs] = [ab[0], ab[1]].map(
x => Array.isArray(x) ? (
x
) : (undefined !== x.type) &&
(x.type.startsWith('Tuple')) ? (
Array.from(x)
) : [x]
),
an = as.length;
return (an === bs.length) ? (
1 < an ? (
traverseList(x => x)(
as.map((_, i) => enumFromTo(as[i])(bs[i]))
)
) : enumFromTo(as[0])(bs[0])
) : [];
};
// snd :: (a, b) -> b const snd = tpl => tpl[1];
// str :: a -> String const str = x => x.toString();
// succ :: Enum a => a -> a
const succ = x =>
1 + x;
// swap :: (a, b) -> (b, a)
const swap = ab =>
// The pair ab with its order reversed.
Tuple(ab[1])(
ab[0]
);
// take :: Int -> [a] -> [a]
// take :: Int -> String -> String
const take = n =>
// The first n elements of a list,
// string of characters, or stream.
xs => xs.slice(0, n);
// traverseList :: (Applicative f) => (a -> f b) -> [a] -> f [b]
const traverseList = f =>
// Collected results of mapping each element
// of a structure to an action, and evaluating
// these actions from left to right.
xs => 0 < xs.length ? (() => {
const
vLast = f(xs.slice(-1)[0]),
t = vLast.type || 'List';
return xs.slice(0, -1).reduceRight(
(ys, x) => liftA2List(cons)(f(x))(ys),
liftA2List(cons)(vLast)([
[]
])
);
})() : [
[]
];
// unlines :: [String] -> String
const unlines = xs =>
// A single string formed by the intercalation
// of a list of strings with the newline character.
xs.join('\n');
// unwords :: [String] -> String
const unwords = xs =>
// A space-separated string derived
// from a list of words.
xs.join(' ');
// zipWith :: (a -> b -> c) -> [a] -> [b] -> [c]
const zipWith = f =>
// A list constructed by zipping with a
// custom function, rather than with the
// default tuple constructor.
xs => ys => {
const
lng = Math.min(length(xs), length(ys)),
vs = take(lng)(ys);
return take(lng)(xs)
.map((x, i) => f(x)(vs[i]));
};
// MAIN --- return main();
})();</lang>
- Output:
Lower: 1 0 0 0 0 1 1 0 0 0 1 2 1 0 0 1 3 3 1 0 1 4 6 4 1 Upper: 1 1 1 1 1 0 1 2 3 4 0 0 1 3 6 0 0 0 1 4 0 0 0 0 1 Symmetric: 1 1 1 1 1 1 2 3 4 5 1 3 6 10 15 1 4 10 20 35 1 5 15 35 70
jq
<lang jq># Generic functions
- Note: 'transpose' is defined in recent versions of jq
def transpose:
if (.[0] | length) == 0 then [] else [map(.[0])] + (map(.[1:]) | transpose) end ;
- Create an m x n matrix with init as the initial value
def matrix(m; n; init):
if m == 0 then []
elif m == 1 then [range(0;n) | init]
elif m > 0 then
matrix(1;n;init) as $row
| [range(0;m) | $row ]
else error("matrix\(m);_;_) invalid")
end ;
- A simple pretty-printer for a 2-d matrix
def pp:
def pad(n): tostring | (n - length) * " " + .;
def row: reduce .[] as $x (""; . + ($x|pad(4)));
reduce .[] as $row (""; . + "\n\($row|row)");</lang>
<lang jq># n is input def pascal_upper:
. as $n
| matrix($n; $n; 0)
| .[0] = [range(0; $n) | 1 ]
| reduce range(1; $n) as $i
(.; reduce range($i; $n) as $j
(.; .[$i][$j] = .[$i-1][$j-1] + .[$i][$j-1]) ) ;
def pascal_lower:
pascal_upper | transpose ;
- n is input
def pascal_symmetric:
. as $n
| matrix($n; $n; 1)
| reduce range(1; $n) as $i
(.; reduce range(1; $n) as $j
(.; .[$i][$j] = .[$i-1][$j] + .[$i][$j-1]) ) ;</lang>
Example: <lang jq>5 | ("\nUpper:", (pascal_upper | pp),
"\nLower:", (pascal_lower | pp), "\nSymmetric:", (pascal_symmetric | pp) )</lang>
- Output:
<lang sh>$ jq -r -n -f Pascal_matrix_generation.jq
Upper:
1 1 1 1 1 0 1 2 3 4 0 0 1 3 6 0 0 0 1 4 0 0 0 0 1
Lower:
1 0 0 0 0 1 1 0 0 0 1 2 1 0 0 1 3 3 1 0 1 4 6 4 1
Symmetric:
1 1 1 1 1 1 2 3 4 5 1 3 6 10 15 1 4 10 20 35 1 5 15 35 70</lang>
Julia
Julia has a built-in binomial function to compute the binomial coefficients, and we can construct the Pascal matrices with this function using list comprehensions:
<lang Julia>julia> [binomial(j,i) for i in 0:4, j in 0:4]
5×5 Array{Int64,2}:
1 1 1 1 1 0 1 2 3 4 0 0 1 3 6 0 0 0 1 4 0 0 0 0 1
julia> [binomial(i,j) for i in 0:4, j in 0:4] 5×5 Array{Int64,2}:
1 0 0 0 0 1 1 0 0 0 1 2 1 0 0 1 3 3 1 0 1 4 6 4 1
julia> [binomial(j+i,i) for i in 0:4, j in 0:4] 5×5 Array{Int64,2}:
1 1 1 1 1 1 2 3 4 5 1 3 6 10 15 1 4 10 20 35 1 5 15 35 70
</lang>
Kotlin
<lang scala>// version 1.1.3
fun binomial(n: Int, k: Int): Int {
if (n < k) return 0
if (n == 0 || k == 0) return 1
val num = (k + 1..n).fold(1) { acc, i -> acc * i }
val den = (2..n - k).fold(1) { acc, i -> acc * i }
return num / den
}
fun pascalUpperTriangular(n: Int) = List(n) { i -> IntArray(n) { j -> binomial(j, i) } }
fun pascalLowerTriangular(n: Int) = List(n) { i -> IntArray(n) { j -> binomial(i, j) } }
fun pascalSymmetric(n: Int) = List(n) { i -> IntArray(n) { j -> binomial(i + j, i) } }
fun printMatrix(title: String, m: List<IntArray>) {
val n = m.size
println(title)
print("[")
for (i in 0 until n) {
if (i > 0) print(" ")
print(m[i].contentToString())
if (i < n - 1) println(",") else println("]\n")
}
}
fun main(args: Array<String>) {
printMatrix("Pascal upper-triangular matrix", pascalUpperTriangular(5))
printMatrix("Pascal lower-triangular matrix", pascalLowerTriangular(5))
printMatrix("Pascal symmetric matrix", pascalSymmetric(5))
}</lang>
- Output:
Pascal upper-triangular matrix [[1, 1, 1, 1, 1], [0, 1, 2, 3, 4], [0, 0, 1, 3, 6], [0, 0, 0, 1, 4], [0, 0, 0, 0, 1]] Pascal lower-triangular matrix [[1, 0, 0, 0, 0], [1, 1, 0, 0, 0], [1, 2, 1, 0, 0], [1, 3, 3, 1, 0], [1, 4, 6, 4, 1]] Pascal symmetric matrix [[1, 1, 1, 1, 1], [1, 2, 3, 4, 5], [1, 3, 6, 10, 15], [1, 4, 10, 20, 35], [1, 5, 15, 35, 70]]
Lua
<lang Lua>function factorial (n)
local f = 1
for i = 2, n do
f = f * i
end
return f
end
function binomial (n, k)
if k > n then return 0 end return factorial(n) / (factorial(k) * factorial(n - k))
end
function pascalMatrix (form, size)
local matrix = {}
for row = 1, size do
matrix[row] = {}
for col = 1, size do
if form == "upper" then
matrix[row][col] = binomial(col - 1, row - 1)
end
if form == "lower" then
matrix[row][col] = binomial(row - 1, col - 1)
end
if form == "symmetric" then
matrix[row][col] = binomial(row + col - 2, col - 1)
end
end
end
matrix.form = form:sub(1, 1):upper() .. form:sub(2, -1)
return matrix
end
function show (mat)
print(mat.form .. ":")
for i = 1, #mat do
for j = 1, #mat[i] do
io.write(mat[i][j] .. "\t")
end
print()
end
print()
end
for _, form in pairs({"upper", "lower", "symmetric"}) do
show(pascalMatrix(form, 5))
end</lang>
- Output:
Upper: 1 1 1 1 1 0 1 2 3 4 0 0 1 3 6 0 0 0 1 4 0 0 0 0 1 Lower: 1 0 0 0 0 1 1 0 0 0 1 2 1 0 0 1 3 3 1 0 1 4 6 4 1 Symmetric: 1 1 1 1 1 1 2 3 4 5 1 3 6 10 15 1 4 10 20 35 1 5 15 35 70
Maple
<lang Maple> PascalUT := proc(n::integer)
local M := Matrix(n,n): local i: local j: M[1,1..n] := 1: for j from 2 to n do for i from 2 to n do M[i,j] := M[i,j-1] + M[i-1,j-1]: end: end: return M:
end proc:
PascalUT(5);
PascalLT := proc(n::integer)
local M := Matrix(n,n): local i: local j: M[1..n,1] := 1: for i from 2 to n do for j from 2 to n do M[i,j] := M[i-1,j] + M[i-1,j-1]: end: end: return M:
end proc:
PascalLT(5);
Pascal := proc(n::integer)
local M := Matrix(n,n): local i: local j: M[1..n,1] := 1: M[1,2..n] := 1: for i from 2 to n do for j from 2 to n do M[i,j] := M[i,j-1] + M[i-1,j]: end: end: return M:
end proc:
Pascal(5);
</lang>
- Output:
[1 1 1 1 1] [ ] [0 1 2 3 4] [ ] [0 0 1 3 6] [ ] [0 0 0 1 4] [ ] [0 0 0 0 1][1 0 0 0 0] [ ] [1 1 0 0 0] [ ] [1 2 1 0 0] [ ] [1 3 3 1 0] [ ] [1 4 6 4 1][1 1 1 1 1] [ ] [1 2 3 4 5] [ ] [1 3 6 10 15] [ ] [1 4 10 20 35] [ ] [1 5 15 35 70]
Mathematica /Wolfram Language
One solution is to generate a symmetric Pascal matrix then use the built in method to compute the upper Pascal matrix. This would be done as follows: <lang Mathematica>symPascal[size_] := NestList[Accumulate, Table[1, {k, size}], size - 1]
upperPascal[size_] := CholeskyDecomposition[symPascal@size]
lowerPascal[size_] := Transpose@CholeskyDecomposition[symPascal@size]
Column[MapThread[
Labeled[Grid[#1@5], #2, Top] &, {{upperPascal, lowerPascal,
symPascal}, {"Upper", "Lower", "Symmetric"}}]]</lang>
- Output:
Upper 1 1 1 1 1 0 1 2 3 4 0 0 1 3 6 0 0 0 1 4 0 0 0 0 1 Lower 1 0 0 0 0 1 1 0 0 0 1 2 1 0 0 1 3 3 1 0 1 4 6 4 1 Symmetric 1 1 1 1 1 1 2 3 4 5 1 3 6 10 15 1 4 10 20 35 1 5 15 35 70
It is also possible to directly compute a lower Pascal matrix as follows: <lang Mathematica>lowerPascal[size_] :=
MatrixExp[
SparseArray[{Band[{2, 1}] -> Range[size - 1]}, {size, size}]]]</lang>
But since the builtin function MatrixExp works by first computing eigenvalues this is likely to be slower for large Pascal matrices
Nim
Using the function “binom” from module “math” of standard library. <lang Nim>import math, sequtils, strutils
type SquareMatrix = seq[seq[Natural]]
func newSquareMatrix(n: Positive): SquareMatrix =
## Create a square matrix. newSeqWith(n, newSeq[Natural](n))
func pascalUpperTriangular(n: Positive): SquareMatrix =
## Create an upper Pascal matrix.
result = newSquareMatrix(n)
for i in 0..<n:
for j in i..<n:
result[i][j] = binom(j, i)
func pascalLowerTriangular(n: Positive): SquareMatrix =
## Create a lower Pascal matrix.
result = newSquareMatrix(n)
for i in 0..<n:
for j in i..<n:
result[j][i] = binom(j, i)
func pascalSymmetric(n: Positive): SquareMatrix =
## Create a symmetric Pascal matrix.
result = newSquareMatrix(n)
for i in 0..<n:
for j in 0..<n:
result[i][j] = binom(i + j, i)
proc print(m: SquareMatrix) =
## Print a square matrix.
let matMax = max(m.mapIt(max(it)))
let length = ($matMax).len
for i in 0..m.high:
echo "| ", m[i].mapIt(($it).align(length)).join(" "), " |"
echo "Upper:" print pascalUpperTriangular(5) echo "\nLower:" print pascalLowerTriangular(5) echo "\nSymmetric:" print pascalSymmetric(5)</lang>
- Output:
Upper: | 1 1 1 1 1 | | 0 1 2 3 4 | | 0 0 1 3 6 | | 0 0 0 1 4 | | 0 0 0 0 1 | Lower: | 1 0 0 0 0 | | 1 1 0 0 0 | | 1 2 1 0 0 | | 1 3 3 1 0 | | 1 4 6 4 1 | Symmetric: | 1 1 1 1 1 | | 1 2 3 4 5 | | 1 3 6 10 15 | | 1 4 10 20 35 | | 1 5 15 35 70 |
PARI/GP
<lang parigp> Pl(n)={matpascal(n-1)} printf("%d",Pl(5)) </lang>
- Output:
[1 0 0 0 0] [1 1 0 0 0] [1 2 1 0 0] [1 3 3 1 0] [1 4 6 4 1]
<lang parigp> Pu(n)={Pl(n)~} printf("%d",Pu(5)) </lang>
- Output:
[1 1 1 1 1] [0 1 2 3 4] [0 0 1 3 6] [0 0 0 1 4] [0 0 0 0 1]
<lang parigp> Ps(n)={matrix(n,n,n,g,binomial(n+g-2,n-1))} printf("%d",Ps(5)) </lang>
- Output:
[1 1 1 1 1] [1 2 3 4 5] [1 3 6 10 15] [1 4 10 20 35] [1 5 15 35 70]
Pascal
<lang pascal>program Pascal_matrix(Output);
const N = 5;
type NxN_Matrix = array[0..N,0..N] of integer;
var PM,PX : NxN_Matrix;
function Pascal_sym(x : integer; p : NxN_Matrix) : NxN_Matrix; var I,J : integer;
begin
for I := 1 to x do
begin
for J := 1 to x do p[I,J] := p[I-1,J]+p[I,J-1]
end;
Pascal_sym := p;
end;
function Pascal_upp(x : integer; p : NxN_Matrix) : NxN_Matrix; var I,J : integer;
begin
for I := 1 to x do
begin
for J := 1 to x do p[I,J] := p[I-1,J-1]+p[I,J-1]
end;
Pascal_upp := p
end;
function Pascal_low(x : integer; p : NxN_Matrix) : NxN_Matrix; var p1,p2 : NxN_Matrix;
I,J : integer;
begin
p1 := Pascal_upp(x,p);
p2 := p1;
for I := 1 to x do
begin
for J := 1 to x do p1[J,I] := p2[I,J]
end;
Pascal_low := p1
end;
procedure PrintMatrix(titel : ansistring; x : integer; p : NxN_Matrix); var I,J : integer;
begin
writeln(titel);
for I := 1 to x do
begin
for J := 1 to x do write(p[I,J]:5);
writeln();
end;
end;
begin
PX[0,0] := 0;
PM[0,0] := 1;
PM := Pascal_upp(N, PM);
PrintMatrix('Upper:', N, PM);
writeln();
PM := PX;
PM[0,0] := 1;
PM := Pascal_low(N, PM);
PrintMatrix('Lower:', N, PM);
writeln();
PM := PX;
PM[1,0] := 1;
PM := Pascal_sym(N, PM);
PrintMatrix('Symmetric', N, PM);
writeln();
readln;
end.</lang>
- Output:
Upper:
1 1 1 1 1
0 1 2 3 4
0 0 1 3 6
0 0 0 1 4
0 0 0 0 1
Lower:
1 0 0 0 0
1 1 0 0 0
1 2 1 0 0
1 3 3 1 0
1 4 6 4 1
Symmetric
1 1 1 1 1
1 2 3 4 5
1 3 6 10 15
1 4 10 20 35
1 5 15 35 70Perl
<lang Perl>#!/usr/bin/perl use warnings; use strict; use feature qw{ say };
sub upper {
my ($i, $j) = @_;
my @m;
for my $x (0 .. $i - 1) {
for my $y (0 .. $j - 1) {
$m[$x][$y] = $x > $y ? 0
: ! $x || $x == $y ? 1
: $m[$x-1][$y-1] + $m[$x][$y-1];
}
}
return \@m
}
sub lower {
my ($i, $j) = @_;
my @m;
for my $x (0 .. $i - 1) {
for my $y (0 .. $j - 1) {
$m[$x][$y] = $x < $y ? 0
: ! $x || $x == $y ? 1
: $m[$x-1][$y-1] + $m[$x-1][$y];
}
}
return \@m
}
sub symmetric {
my ($i, $j) = @_;
my @m;
for my $x (0 .. $i - 1) {
for my $y (0 .. $j - 1) {
$m[$x][$y] = ! $x || ! $y ? 1
: $m[$x-1][$y] + $m[$x][$y-1];
}
}
return \@m
}
sub pretty {
my $m = shift;
for my $row (@$m) {
say join ', ', @$row;
}
}
pretty(upper(5, 5));
say '-' x 14;
pretty(lower(5, 5));
say '-' x 14;
pretty(symmetric(5, 5));</lang>
- Output:
1, 1, 1, 1, 1 0, 1, 2, 3, 4 0, 0, 1, 3, 6 0, 0, 0, 1, 4 0, 0, 0, 0, 1 -------------- 1, 0, 0, 0, 0 1, 1, 0, 0, 0 1, 2, 1, 0, 0 1, 3, 3, 1, 0 1, 4, 6, 4, 1 -------------- 1, 1, 1, 1, 1 1, 2, 3, 4, 5 1, 3, 6, 10, 15 1, 4, 10, 20, 35 1, 5, 15, 35, 70
Phix
function pascal_upper(integer n) sequence res = repeat(repeat(0,n),n) res[1] = repeat(1,n) for i=2 to n do for j=2 to i do res[j,i] = res[j,i-1]+res[j-1,i-1] end for end for return res end function function pascal_lower(integer n) sequence res = repeat(repeat(0,n),n) for i=1 to n do res[i,1] = 1 end for for i=2 to n do for j=2 to i do res[i,j] = res[i-1,j]+res[i-1,j-1] end for end for return res end function function pascal_symmetric(integer n) sequence res = repeat(repeat(0,n),n) for i=1 to n do res[i,1] = 1 res[1,i] = 1 end for for i=2 to n do for j = 2 to n do res[i,j] = res[i-1,j]+res[i,j-1] end for end for return res end function ppOpt({pp_Nest,1,pp_IntCh,false,pp_IntFmt,"%2d"}) puts(1,"=== Pascal upper matrix ===\n") pp(pascal_upper(5)) puts(1,"=== Pascal lower matrix ===\n") pp(pascal_lower(5)) puts(1,"=== Pascal symmetrical matrix ===\n") pp(pascal_symmetric(5))
- Output:
=== Pascal upper matrix ===
{{ 1, 1, 1, 1, 1},
{ 0, 1, 2, 3, 4},
{ 0, 0, 1, 3, 6},
{ 0, 0, 0, 1, 4},
{ 0, 0, 0, 0, 1}}
=== Pascal lower matrix ===
{{ 1, 0, 0, 0, 0},
{ 1, 1, 0, 0, 0},
{ 1, 2, 1, 0, 0},
{ 1, 3, 3, 1, 0},
{ 1, 4, 6, 4, 1}}
=== Pascal symmetrical matrix ===
{{ 1, 1, 1, 1, 1},
{ 1, 2, 3, 4, 5},
{ 1, 3, 6,10,15},
{ 1, 4,10,20,35},
{ 1, 5,15,35,70}}
PicoLisp
<lang PicoLisp>(setq
Low '(A B) Upp '(B A) Sym '((+ A B) A) )
(de binomial (N K)
(let f
'((N)
(if (=0 N) 1 (apply * (range 1 N))) )
(if (> K N)
0
(/
(f N)
(* (f (- N K)) (f K)) ) ) ) )
(de pascal (N Z)
(for Lst
(mapcar
'((A)
(mapcar
'((B) (apply binomial (mapcar eval Z)))
(range 0 N) ) )
(range 0 N) )
(for L Lst
(prin (align 2 L) " ") )
(prinl) )
(prinl) )
(pascal 4 Low) (pascal 4 Upp) (pascal 4 Sym)</lang>
- Output:
1 0 0 0 0 1 1 0 0 0 1 2 1 0 0 1 3 3 1 0 1 4 6 4 1 1 1 1 1 1 0 1 2 3 4 0 0 1 3 6 0 0 0 1 4 0 0 0 0 1 1 1 1 1 1 1 2 3 4 5 1 3 6 10 15 1 4 10 20 35 1 5 15 35 70
PL/I
<lang pli> PASCAL_MATRIX: PROCEDURE OPTIONS (MAIN); /* derived from Fortran version 18 Decenber 2021 */
pascal_lower: procedure(a);
declare a(*,*) fixed binary;
declare (n, i, j) fixed binary;
n = hbound(a,1);
a = 0;
a(*, 1) = 1;
do i = 2 to n;
do j = 2 to i;
a(i, j) = a(i - 1, j) + a(i - 1, j - 1);
end;
end;
end pascal_lower;
pascal_upper: procedure(a);
declare a(*,*) fixed binary;
declare (n, i, j) fixed binary;
n = hbound(a,1);
a = 0;
a(1, *) = 1;
do i = 2 to n;
do j = 2 to i;
a(j, i) = a(j, i - 1) + a(j - 1, i - 1);
end;
end;
end pascal_upper;
pascal_symmetric: procedure(a);
declare a(*,*) fixed binary;
declare (n, i, j) fixed binary;
n = hbound(a,1);
a = 0;
a(*, 1) = 1;
a(1, *) = 1;
do i = 2 to n;
do j = 2 to n;
a(i, j) = a(i - 1, j) + a(i, j - 1);
end;
end;
end pascal_symmetric;
declare n fixed binary;
put ('Size of matrix?');
get (n);
begin;
declare a(n, n) fixed binary;
put skip list ('Lower Pascal Matrix');
call pascal_lower(a);
put edit (a) (skip, (n) f(3) );
put skip list ('Upper Pascal Matrix');
call pascal_upper(a);
put edit (a) (skip, (n) f(3) );
put skip list ('Symmetric Pascal Matrix');
call pascal_symmetric(a);
put edit (a) (skip, (n) f(3) );
end;
end PASCAL_MATRIX; </lang>
- Output:
Size of matrix? Lower Pascal Matrix 1 0 0 0 0 1 1 0 0 0 1 2 1 0 0 1 3 3 1 0 1 4 6 4 1 Upper Pascal Matrix 1 1 1 1 1 0 1 2 3 4 0 0 1 3 6 0 0 0 1 4 0 0 0 0 1 Symmetric Pascal Matrix 1 1 1 1 1 1 2 3 4 5 1 3 6 10 15 1 4 10 20 35 1 5 15 35 70
<lang pli>*process source attributes xref or(!);
pat: Proc Options(main); Dcl (HBOUND,MAX,RIGHT) Builtin; Dcl SYSPRINT Print; Dcl N Bin Fixed(31) Init(5); Dcl pd Char(500) Var; Dcl fact(0:10) Bin Fixed(31); Dcl pt(0:500) Bin Fixed(31); Call mk_fact(fact);
Call Pascal(n,'U',pt); Call show('Pascal upper triangular matrix');
Call Pascal(n,'L',pt); Call show('Pascal lower triangular matrix');
Call Pascal(n,'S',pt); Call show('Pascal symmetric matrix' );
Pascal: proc(n,which,dd);
Dcl n Bin Fixed(31);
Dcl which Char(1);
Dcl (i,j,k) Bin Fixed(31);
Dcl dd(0:500) Bin Fixed(31);
k=0;
dd(0)=0;
do i=0 To n-1;
Do j=0 To n-1;
k+=1;
Select(which);
When('U') dd(k)=comb((j), (i));
When('L') dd(k)=comb((i), (j));
When('S') dd(k)=comb((i+j),(i));
Otherwise;
End;
dd(0)=max(dd(0),dd(k));
End;
End;
End;
mk_fact: Proc(f); Dcl f(0:*) Bin Fixed(31); Dcl i Bin Fixed(31); f(0)=1; Do i=1 To hbound(f); f(i)=f(i-1)*i; End; End;
comb: proc(x,y) Returns(pic'z9');
Dcl (x,y) Bin Fixed(31);
Dcl (j,z) Bin Fixed(31);
Dcl res Pic'Z9';
Select;
When(x=y) res=1;
When(y>x) res=0;
Otherwise Do;
If x-y<y then
y=x-y;
z=1;
do j=x-y+1 to x;
z=z*j;
End;
res=z/fact(y);
End;
End;
Return(res);
End;
show: Proc(head);
Dcl head Char(*);
Dcl (n,r,c,pl) Bin Fixed(31) Init(0);
Dcl row Char(50) Var;
Dcl p Pic'z9';
If pt(0)<10 Then pl=1;
Else pl=2;
Dcl sep(5) Char(1) Init((4)(1)',',']');
Put Edit(' ',head)(Skip,a);
do r=1 To 5;
if r=1 then row='[[';
else row=' [';
do c=1 To 5;
n+=1;
p=pt(n);
row=row!!right(p,pl)!!sep(c);
End;
Put Edit(row)(Skip,a);
End;
Put Edit(']')(A);
End;
End;</lang>
- Output:
Pascal upper triangular matrix [[1,1,1,1,1] [0,1,2,3,4] [0,0,1,3,6] [0,0,0,1,4] [0,0,0,0,1]] Pascal lower triangular matrix [[1,0,0,0,0] [1,1,0,0,0] [1,2,1,0,0] [1,3,3,1,0] [1,4,6,4,1]] Pascal symmetric matrix [[ 1, 1, 1, 1, 1] [ 1, 2, 3, 4, 5] [ 1, 3, 6,10,15] [ 1, 4,10,20,35] [ 1, 5,15,35,70]]
PureBasic
<lang PureBasic>EnableExplicit Define.i x=5, I, J
Macro Print_Pascal_matrix(typ)
PrintN(typ)
For I=1 To x
For J=1 To x : Print(RSet(Str(p(I,J)),3," ")+Space(3)) : Next
PrintN("")
Next
Print(~"\n\n")
EndMacro
Procedure Pascal_sym(n.i,Array p.i(2))
Define.i I,J p(1,0)=1 For I=1 To n For J=1 To n : p(I,J)=p(I-1,J)+p(I,J-1) : Next Next
EndProcedure
Procedure Pascal_upp(n.i,Array p.i(2))
Define.i I,J p(0,0)=1 For I=1 To n For J=1 To n : p(I,J)=p(I-1,J-1)+p(I,J-1) : Next Next
EndProcedure
Procedure Pascal_low(n.i,Array p.i(2))
Define.i I,J Pascal_upp(n,p()) Dim p2.i(n,n) CopyArray(p(),p2()) For I=1 To n For J=1 To n : Swap p(J,I),p2(I,J) : Next Next
EndProcedure
OpenConsole()
Dim p.i(x,x) Pascal_upp(x,p()) Print_Pascal_matrix("Upper:")
Dim p.i(x,x) Pascal_low(x,p()) Print_Pascal_matrix("Lower:")
Dim p.i(x,x) Pascal_sym(x,p()) Print_Pascal_matrix("Symmetric:")
Input() End</lang>
- Output:
Upper: 1 1 1 1 1 0 1 2 3 4 0 0 1 3 6 0 0 0 1 4 0 0 0 0 1 Lower: 1 0 0 0 0 1 1 0 0 0 1 2 1 0 0 1 3 3 1 0 1 4 6 4 1 Symmetric: 1 1 1 1 1 1 2 3 4 5 1 3 6 10 15 1 4 10 20 35 1 5 15 35 70
Python
Python: Procedural
Summing adjacent values: <lang python>from pprint import pprint as pp
def pascal_upp(n):
s = [[0] * n for _ in range(n)]
s[0] = [1] * n
for i in range(1, n):
for j in range(i, n):
s[i][j] = s[i-1][j-1] + s[i][j-1]
return s
def pascal_low(n):
# transpose of pascal_upp(n) return [list(x) for x in zip(*pascal_upp(n))]
def pascal_sym(n):
s = [[1] * n for _ in range(n)]
for i in range(1, n):
for j in range(1, n):
s[i][j] = s[i-1][j] + s[i][j-1]
return s
if __name__ == "__main__":
n = 5
print("\nUpper:")
pp(pascal_upp(n))
print("\nLower:")
pp(pascal_low(n))
print("\nSymmetric:")
pp(pascal_sym(n))</lang>
- Output:
Upper: [[1, 1, 1, 1, 1], [0, 1, 2, 3, 4], [0, 0, 1, 3, 6], [0, 0, 0, 1, 4], [0, 0, 0, 0, 1]] Lower: [[1, 0, 0, 0, 0], [1, 1, 0, 0, 0], [1, 2, 1, 0, 0], [1, 3, 3, 1, 0], [1, 4, 6, 4, 1]] Symmetric: [[1, 1, 1, 1, 1], [1, 2, 3, 4, 5], [1, 3, 6, 10, 15], [1, 4, 10, 20, 35], [1, 5, 15, 35, 70]]
Using a binomial coefficient generator function:
<lang python>def binomialCoeff(n, k):
result = 1
for i in range(1, k+1):
result = result * (n-i+1) // i
return result
def pascal_upp(n):
return [[binomialCoeff(j, i) for j in range(n)] for i in range(n)]
def pascal_low(n):
return [[binomialCoeff(i, j) for j in range(n)] for i in range(n)]
def pascal_sym(n):
return [[binomialCoeff(i+j, i) for j in range(n)] for i in range(n)]</lang>
- Output:
(As above)
Python: Functional
Defining binomial coefficients in terms of reduce: <lang python>Pascal matrix generation
from functools import reduce from itertools import chain from operator import add
- pascalMatrix :: Int -> ((Int, Int) -> (Int, Int)) -> Int
def pascalMatrix(n):
Pascal S-, L-, or U- matrix of order n.
return lambda f: chunksOf(n)(list(map(
compose(binomialCoefficent, f),
tupleRange((0, 0), (n, n))
)))
- binomialCoefficent :: (Int, Int) -> Int
def binomialCoefficent(nk):
The binomial coefficient of the tuple (n, k). n, k = nk
def go(a, x):
return a * (n - x + 1) // x
return reduce(go, enumFromTo(1)(k), 1)
- --------------------------TEST---------------------------
- main :: IO ()
def main():
Pascal S-, L-, and U- matrices of order 5.
order = 5
for k, f in [
('Symmetric', lambda ab: (add(*ab), ab[1])),
('Lower', identity),
('Upper', swap)
]:
print(k + ':')
print(showMatrix(
pascalMatrix(order)(f)
))
print()
- --------------------REUSABLE GENERICS--------------------
- chunksOf :: Int -> [a] -> a
def chunksOf(n):
A series of lists of length n, subdividing the
contents of xs. Where the length of xs is not evenly
divible, the final list will be shorter than n.
return lambda xs: reduce(
lambda a, i: a + [xs[i:n + i]],
range(0, len(xs), n), []
) if 0 < n else []
- compose :: ((a -> a), ...) -> (a -> a)
def compose(*fs):
Composition, from right to left,
of a series of functions.
return lambda x: reduce(
lambda a, f: f(a),
fs[::-1], x
)
- enumFromTo :: Int -> Int -> [Int]
def enumFromTo(m):
Enumeration of integer values [m..n] return lambda n: range(m, 1 + n)
- identity :: a -> a
def identity(x):
The identity function. return x
- showMatrix :: Int -> String
def showMatrix(xs):
String representation of xs
as a matrix.
def go():
rows = [[str(x) for x in row] for row in xs]
w = max(map(len, chain.from_iterable(rows)))
return unlines(
unwords(k.rjust(w, ' ') for k in row)
for row in rows
)
return go() if xs else
- swap :: (a, b) -> (b, a)
def swap(tpl):
The swapped components of a pair. return tpl[1], tpl[0]
- tupleRange :: (Int, Int) -> (Int, Int) -> [(Int, Int)]
def tupleRange(lowerTuple, upperTuple):
Range of (Int, Int) tuples from
lowerTuple to upperTuple.
l1, l2 = lowerTuple
u1, u2 = upperTuple
return [
(i1, i2) for i1 in range(l1, u1)
for i2 in range(l2, u2)
]
- unlines :: [String] -> String
def unlines(xs):
A single string formed by the intercalation
of a list of strings with the newline character.
return '\n'.join(xs)
- unwords :: [String] -> String
def unwords(xs):
A space-separated string derived
from a list of words.
return ' '.join(xs)
- MAIN ---
if __name__ == '__main__':
main()</lang>
- Output:
Symmetric: 1 1 1 1 1 1 2 3 4 5 1 3 6 10 15 1 4 10 20 35 1 5 15 35 70 Lower: 1 0 0 0 0 1 1 0 0 0 1 2 1 0 0 1 3 3 1 0 1 4 6 4 1 Upper: 1 1 1 1 1 0 1 2 3 4 0 0 1 3 6 0 0 0 1 4 0 0 0 0 1
R
<lang r>lower.pascal <- function(n) {
a <- matrix(0, n, n)
a[, 1] <- 1
if (n > 1) {
for (i in 2:n) {
j <- 2:i
a[i, j] <- a[i - 1, j - 1] + a[i - 1, j]
}
}
a
}
- Alternate version
lower.pascal.alt <- function(n) {
a <- matrix(0, n, n)
a[, 1] <- 1
if (n > 1) {
for (j in 2:n) {
i <- j:n
a[i, j] <- cumsum(a[i - 1, j - 1])
}
}
a
}
- While it's possible to modify lower.pascal to get the upper matrix,
- here we simply transpose the lower one.
upper.pascal <- function(n) t(lower.pascal(n))
symm.pascal <- function(n) {
a <- matrix(0, n, n)
a[, 1] <- 1
for (i in 2:n) {
a[, i] <- cumsum(a[, i - 1])
}
a
}</lang>
The results follow
<lang r>> lower.pascal(5)
[,1] [,2] [,3] [,4] [,5]
[1,] 1 0 0 0 0 [2,] 1 1 0 0 0 [3,] 1 2 1 0 0 [4,] 1 3 3 1 0 [5,] 1 4 6 4 1 > lower.pascal.alt(5)
[,1] [,2] [,3] [,4] [,5]
[1,] 1 0 0 0 0 [2,] 1 1 0 0 0 [3,] 1 2 1 0 0 [4,] 1 3 3 1 0 [5,] 1 4 6 4 1 > upper.pascal(5)
[,1] [,2] [,3] [,4] [,5]
[1,] 1 1 1 1 1 [2,] 0 1 2 3 4 [3,] 0 0 1 3 6 [4,] 0 0 0 1 4 [5,] 0 0 0 0 1 > symm.pascal(5)
[,1] [,2] [,3] [,4] [,5]
[1,] 1 1 1 1 1 [2,] 1 2 3 4 5 [3,] 1 3 6 10 15 [4,] 1 4 10 20 35 [5,] 1 5 15 35 70</lang>
Racket
<lang racket>#lang racket (require math/number-theory)
(define (pascal-upper-matrix n)
(for/list ((i n)) (for/list ((j n)) (j . binomial . i))))
(define (pascal-lower-matrix n)
(for/list ((i n)) (for/list ((j n)) (i . binomial . j))))
(define (pascal-symmetric-matrix n)
(for/list ((i n)) (for/list ((j n)) ((+ i j) . binomial . j))))
(define (matrix->string m)
(define col-width
(for*/fold ((rv 1)) ((r m) (c r))
(if (zero? c) rv (max rv (+ 1 (order-of-magnitude c))))))
(string-append
(string-join
(for/list ((r m))
(string-join (map (λ (c) (~a #:width col-width #:align 'right c)) r) " ")) "\n")
"\n"))
(printf "Upper:~%~a~%" (matrix->string (pascal-upper-matrix 5))) (printf "Lower:~%~a~%" (matrix->string (pascal-lower-matrix 5))) (printf "Symmetric:~%~a~%" (matrix->string (pascal-symmetric-matrix 5)))</lang>
- Output:
Upper: 1 1 1 1 1 0 1 2 3 4 0 0 1 3 6 0 0 0 1 4 0 0 0 0 1 Lower: 1 0 0 0 0 1 1 0 0 0 1 2 1 0 0 1 3 3 1 0 1 4 6 4 1 Symmetric: 1 1 1 1 1 1 2 3 4 5 1 3 6 10 15 1 4 10 20 35 1 5 15 35 70
Raku
(formerly Perl 6)
Here is a rather more general solution than required. The grow-matrix function will grow any N by N matrix into an N+1 x N+1 matrix, using any function of the three leftward/upward neighbors, here labelled "West", "North", and "Northwest". We then define three iterator functions that can grow Pascal matrices, and use those iterators to define three constants, each of which is an infinite sequence of ever-larger Pascal matrices. Normal subscripting then pulls out the ones of the specified size. <lang perl6># Extend a matrix in 2 dimensions based on 3 neighbors. sub grow-matrix(@matrix, &func) {
my $n = @matrix.shape eq '*' ?? 1 !! @matrix.shape[0];
my @m[$n+1;$n+1];
for ^$n X ^$n -> ($i, $j) {
@m[$i;$j] = @matrix[$i;$j];
}
- West North NorthWest
@m[$n; 0] = func( 0, @m[$n-1;0], 0 ); @m[ 0;$n] = func( @m[0;$n-1], 0, 0 ); @m[$_;$n] = func( @m[$_;$n-1], @m[$_-1;$n], @m[$_-1;$n-1]) for 1 ..^ $n; @m[$n;$_] = func( @m[$n;$_-1], @m[$n-1;$_], @m[$n-1;$_-1]) for 1 .. $n; @m;
}
- I am but mad north-northwest...
sub madd-n-nw(@m) { grow-matrix @m, -> $w, $n, $nw { $n + $nw } } sub madd-w-nw(@m) { grow-matrix @m, -> $w, $n, $nw { $w + $nw } } sub madd-w-n (@m) { grow-matrix @m, -> $w, $n, $nw { $w + $n } }
- Define 3 infinite sequences of Pascal matrices.
constant upper-tri = [1], &madd-w-nw ... *; constant lower-tri = [1], &madd-n-nw ... *; constant symmetric = [1], &madd-w-n ... *;
show_m upper-tri[4]; show_m lower-tri[4]; show_m symmetric[4];
sub show_m (@m) { my \n = @m.shape[0]; for ^n X ^n -> (\i, \j) {
print @m[i;j].fmt("%{1+max(@m).chars}d");
print "\n" if j+1 eq n;
} say ; }</lang>
- Output:
1 1 1 1 1 0 1 2 3 4 0 0 1 3 6 0 0 0 1 4 0 0 0 0 1 1 0 0 0 0 1 1 0 0 0 1 2 1 0 0 1 3 3 1 0 1 4 6 4 1 1 1 1 1 1 1 2 3 4 5 1 3 6 10 15 1 4 10 20 35 1 5 15 35 70
REXX
separate generation
Commentary: 1/3 of the REXX program deals with the displaying of the matrix. <lang rexx>/*REXX program generates and displays three forms of an NxN Pascal matrix. */ numeric digits 50 /*be able to calculate huge factorials.*/ parse arg N . /*obtain the optional matrix size (N).*/ if N== | N=="," then N= 5 /*Not specified? Then use the default.*/
call show N, upp(N), 'Pascal upper triangular matrix'
call show N, low(N), 'Pascal lower triangular matrix'
call show N, sym(N), 'Pascal symmetric matrix'
exit /*stick a fork in it, we're all done. */ /*──────────────────────────────────────────────────────────────────────────────────────*/ upp: procedure; parse arg N; $= /*gen Pascal upper triangular matrix. */
do i=0 for N; do j=0 for N; $=$ comb(j, i); end; end; return $
/*──────────────────────────────────────────────────────────────────────────────────────*/ low: procedure; parse arg N; $= /*gen Pascal lower triangular matrix. */
do i=0 for N; do j=0 for N; $=$ comb(i, j); end; end; return $
/*──────────────────────────────────────────────────────────────────────────────────────*/ sym: procedure; parse arg N; $= /*generate Pascal symmetric matrix. */
do i=0 for N; do j=0 for N; $=$ comb(i+j, i); end; end; return $
/*──────────────────────────────────────────────────────────────────────────────────────*/ !: procedure; parse arg x; !=1; do j=2 to x; != !*j; end; return ! /*──────────────────────────────────────────────────────────────────────────────────────*/ comb: procedure; parse arg x,y; if x=y then return 1 /* {=} case.*/
if y>x then return 0 /* {>} case.*/
if x-y<y then y= x-y; _= 1; do j=x-y+1 to x; _= _*j; end; return _ / !(y)
/*──────────────────────────────────────────────────────────────────────────────────────*/ show: procedure; parse arg s,@; w=0; #=0 /*get args. */
do x=1 for s**2; w= max(w, 1 + length( word(@,x) ) ); end
say; say center( arg(3), 50, '─') /*show title*/
do r=1 for s; if r==1 then $= '[[' /*row 1 */
else $= ' [' /*rows 2 N*/
do c=1 for s; #= #+1; e= (c==s) /*e ≡ "end".*/
$=$ || right( word(@, #), w) || left(',', \e) || left("]", e)
end /*c*/ /* [↑] row.*/
say $ || left(',', r\==s)left("]", r==s) /*show row. */
end /*r*/
return</lang>
- output when using the default input:
──────────Pascal upper triangular matrix────────── [[ 1, 1, 1, 1, 1], [ 0, 1, 2, 3, 4], [ 0, 0, 1, 3, 6], [ 0, 0, 0, 1, 4], [ 0, 0, 0, 0, 1]] ──────────Pascal lower triangular matrix────────── [[ 1, 0, 0, 0, 0], [ 1, 1, 0, 0, 0], [ 1, 2, 1, 0, 0], [ 1, 3, 3, 1, 0], [ 1, 4, 6, 4, 1]] ─────────────Pascal symmetric matrix────────────── [[ 1, 1, 1, 1, 1], [ 1, 2, 3, 4, 5], [ 1, 3, 6, 10, 15], [ 1, 4, 10, 20, 35], [ 1, 5, 15, 35, 70]]
consolidated generation
This REXX version uses a consolidated generation subroutine, even though this Rosetta Code implies to use functions (instead of a single function). <lang rexx>/*REXX program generates and displays three forms of an NxN Pascal matrix. */ numeric digits 50 /*be able to calculate huge factorials.*/ parse arg N . /*obtain the optional matrix size (N).*/ if N== | N=="," then N= 5 /*Not specified? Then use the default.*/
call show N, Pmat(N, 'upper'), 'Pascal upper triangular matrix'
call show N, Pmat(N, 'lower'), 'Pascal lower triangular matrix'
call show N, Pmat(N, 'sym') , 'Pascal symmetric matrix'
exit /*stick a fork in it, we're all done. */ /*──────────────────────────────────────────────────────────────────────────────────────*/ Pmat: procedure; parse arg N; $= /*generate a format of a Pascal matrix.*/
arg , ? /*get uppercase version of the 2nd arg.*/
do i=0 for N; do j=0 for N /*pick a format to use [↓] */
if abbrev('UPPER' , ?, 1) then $= $ comb(j , i)
if abbrev('LOWER' , ?, 1) then $= $ comb(i , j)
if abbrev('SYMMETRICAL', ?, 1) then $= $ comb(i+j, j)
end /*j*/ /* ↑ */
end /*i*/ /* │ */
return $ /* └──min. abbreviation is 1 char.*/
/*──────────────────────────────────────────────────────────────────────────────────────*/ !: procedure; parse arg x; !=1; do j=2 to x; != ! * j; end; return ! /*──────────────────────────────────────────────────────────────────────────────────────*/ comb: procedure; parse arg x,y; if x=y then return 1 /* {=} case.*/
if y>x then return 0 /* {>} case.*/
if x-y<y then y=x-y; _= 1; do j=x-y+1 to x; _= _ * j; end; return _ / !(y)
/*──────────────────────────────────────────────────────────────────────────────────────*/ show: procedure; parse arg s,@; w=0; #=0 /*get args. */
do x=1 for s**2; w=max(w,1+length(word(@,x))); end
say; say center( arg(3), 50, '─') /*show title*/
do r=1 for s; if r==1 then $= '[[' /*row 1 */
else $= ' [' /*rows 2 N*/
do c=1 for s; #= # + 1; e= (c==s) /*e ≡ "end".*/
$=$ || right( word(@, #), w) || left(', ',\e) || left("]", e)
end /*c*/ /* [↑] row.*/
say $ || left(',', r\==s)left(']', r==s) /*show row. */
end /*r*/
return</lang>
- output is identical to the 1st REXX version.
Ring
<lang ring>
- Project : Pascal matrix generation
load "stdlib.ring" res = newlist(5,5)
see "=== Pascal upper matrix ===" + nl result = pascalupper(5) showarray(result)
see nl + "=== Pascal lower matrix ===" + nl result = pascallower(5) showarray(result)
see nl + "=== Pascal symmetrical matrix ===" + nl result = pascalsymmetric(5) showarray(result)
func pascalupper(n)
for m=1 to n
for p=1 to n
res[m][p] = 0
next
next
for p=1 to n
res[1][p] = 1
next
for i=2 to n
for j=2 to i
res[j][i] = res[j][i-1]+res[j-1][i-1]
end
end
return res
func pascallower(n)
for m=1 to n
for p=1 to n
res[m][p] = 0
next
next
for p=1 to n
res[p][1] = 1
next
for i=2 to n
for j=2 to i
res[i][j] = res[i-1][j]+res[i-1][j-1]
next
next
return res
func pascalsymmetric(n)
for m=1 to n
for p=1 to n
res[m][p] = 0
next
next
for p=1 to n
res[p][1] = 1
res[1][p] = 1
next
for i=2 to n
for j = 2 to n
res[i][j] = res[i-1][j]+res[i][j-1]
next
next
return res
func showarray(result)
for n=1 to 5
for m=1 to 5
see "" + result[n][m] + " "
next
see nl
next
</lang> Output:
=== Pascal upper matrix === 1 1 1 1 1 0 1 2 3 4 0 0 1 3 6 0 0 0 1 4 0 0 0 0 1 === Pascal lower matrix === 1 0 0 0 0 1 1 0 0 0 1 2 1 0 0 1 3 3 1 0 1 4 6 4 1 === Pascal symmetrical matrix === 1 1 1 1 1 1 2 3 4 5 1 3 6 10 15 1 4 10 20 35 1 5 15 35 70
Ruby
Summing adjacent values: <lang ruby>#Upper, lower, and symetric Pascal Matrix - Nigel Galloway: May 3rd., 21015 require 'pp'
ng = (g = 0..4).collect{[]} g.each{|i| g.each{|j| ng[i][j] = i==0 ? 1 : j<i ? 0 : ng[i-1][j-1]+ng[i][j-1]}} pp ng; puts g.each{|i| g.each{|j| ng[i][j] = j==0 ? 1 : i<j ? 0 : ng[i-1][j-1]+ng[i-1][j]}} pp ng; puts g.each{|i| g.each{|j| ng[i][j] = (i==0 or j==0) ? 1 : ng[i-1][j ]+ng[i][j-1]}} pp ng</lang>
- Output:
[[1, 1, 1, 1, 1], [0, 1, 2, 3, 4], [0, 0, 1, 3, 6], [0, 0, 0, 1, 4], [0, 0, 0, 0, 1]] [[1, 0, 0, 0, 0], [1, 1, 0, 0, 0], [1, 2, 1, 0, 0], [1, 3, 3, 1, 0], [1, 4, 6, 4, 1]] [[1, 1, 1, 1, 1], [1, 2, 3, 4, 5], [1, 3, 6, 10, 15], [1, 4, 10, 20, 35], [1, 5, 15, 35, 70]]
Binomial coefficient:
<lang ruby>require 'pp'
def binomial_coeff(n,k) (1..k).inject(1){|res,i| res * (n-i+1) / i} end
def pascal_upper(n) (0...n).map{|i| (0...n).map{|j| binomial_coeff(j,i)}} end def pascal_lower(n) (0...n).map{|i| (0...n).map{|j| binomial_coeff(i,j)}} end def pascal_symmetric(n) (0...n).map{|i| (0...n).map{|j| binomial_coeff(i+j,j)}} end
puts "Pascal upper-triangular matrix:" pp pascal_upper(5)
puts "\nPascal lower-triangular matrix:" pp pascal_lower(5)
puts "\nPascal symmetric matrix:" pp pascal_symmetric(5)</lang>
- Output:
Pascal upper-triangular matrix: [[1, 1, 1, 1, 1], [0, 1, 2, 3, 4], [0, 0, 1, 3, 6], [0, 0, 0, 1, 4], [0, 0, 0, 0, 1]] Pascal lower-triangular matrix: [[1, 0, 0, 0, 0], [1, 1, 0, 0, 0], [1, 2, 1, 0, 0], [1, 3, 3, 1, 0], [1, 4, 6, 4, 1]] Pascal symmetric matrix: [[1, 1, 1, 1, 1], [1, 2, 3, 4, 5], [1, 3, 6, 10, 15], [1, 4, 10, 20, 35], [1, 5, 15, 35, 70]]
Scala
<lang scala>//Pascal Matrix Generator
object pascal{ def main( args:Array[String] ){
println("Enter the order of matrix") val n = scala.io.StdIn.readInt()
var F = new Factorial()
var mx = Array.ofDim[Int](n,n)
for( i <- 0 to (n-1); j <- 0 to (n-1) ){
if( i>=j ){ //iCj mx(i)(j) = F.fact(i) / ( ( F.fact(j) )*( F.fact(i-j) ) ) } }
println("iCj:") for( i <- 0 to (n-1) ){ //iCj print for( j <- 0 to (n-1) ){ print( mx(i)(j)+" " ) } println("") }
println("jCi:") for( i <- 0 to (n-1) ){ //jCi print for( j <- 0 to (n-1) ){ print( mx(j)(i)+" " ) } println("") }
//(i+j)C j for( i <- 0 to (n-1); j <- 0 to (n-1) ){
mx(i)(j) = F.fact(i+j) / ( ( F.fact(j) )*( F.fact(i) ) ) } //print (i+j)Cj println("(i+j)Cj:") for( i <- 0 to (n-1) ){ for( j <- 0 to (n-1) ){ print( mx(i)(j)+" " ) } println("") }
} }
class Factorial(){
def fact( a:Int ): Int = {
var b:Int = 1
for( i <- 2 to a ){ b = b*i } return b } } </lang>
Scheme
Using SRFI-25:
<lang scheme>(import (srfi 25))
(define-syntax dotimes
(syntax-rules ()
((_ (i n) body ...)
(do ((i 0 (+ i 1)))
((>= i n))
body ...))))
(define (pascal-upper n)
(let ((p (make-array (shape 0 n 0 n) 0)))
(dotimes (i n)
(array-set! p 0 i 1))
(dotimes (i (- n 1))
(dotimes (j (- n 1))
(array-set! p (+ 1 i) (+ 1 j)
(+ (array-ref p i j)
(array-ref p (+ 1 i) j)))))
p))
(define (pascal-lower n)
(let ((p (make-array (shape 0 n 0 n) 0)))
(dotimes (i n)
(array-set! p i 0 1))
(dotimes (i (- n 1))
(dotimes (j (- n 1))
(array-set! p (+ 1 i) (+ 1 j)
(+ (array-ref p i j)
(array-ref p i (+ 1 j))))))
p))
(define (pascal-symmetric n)
(let ((p (make-array (shape 0 n 0 n) 0)))
(dotimes (i n)
(array-set! p i 0 1)
(array-set! p 0 i 1))
(dotimes (i (- n 1))
(dotimes (j (- n 1))
(array-set! p (+ 1 i) (+ 1 j)
(+ (array-ref p (+ 1 i) j)
(array-ref p i (+ 1 j))))))
p))
(define (print-array a)
(let ((r (array-end a 0))
(c (array-end a 1)))
(dotimes (row (- r 1))
(dotimes (col (- c 1))
(display (array-ref a row col))
(display #\space))
(newline))))</lang>
<lang scheme>(print-array (pascal-upper 6))</lang>
- Output:
1 1 1 1 1 0 1 2 3 4 0 0 1 3 6 0 0 0 1 4 0 0 0 0 1
<lang scheme>(print-array (pascal-lower 6))</lang>
- Output:
1 0 0 0 0 1 1 0 0 0 1 2 1 0 0 1 3 3 1 0 1 4 6 4 1
<lang scheme>(print-array (pascal-symmetric 6))</lang>
- Output:
1 1 1 1 1 1 2 3 4 5 1 3 6 10 15 1 4 10 20 35 1 5 15 35 70
Sidef
<lang ruby>func grow_matrix(matrix, callback) {
var m = matrix var s = m.len m[s][0] = callback(0, m[s-1][0], 0) m[0][s] = callback(m[0][s-1], 0, 0)
- Output:
1 1 1 1 1 0 1 2 3 4 0 0 1 3 6 0 0 0 1 4 0 0 0 0 1 1 0 0 0 0 1 1 0 0 0 1 2 1 0 0 1 3 3 1 0 1 4 6 4 1 1 1 1 1 1 1 2 3 4 5 1 3 6 10 15 1 4 10 20 35 1 5 15 35 70
Stata
Here are variants for the lower matrix.
<lang stata>mata function pascal1(n) { return(comb(J(1,n,0::n-1),J(n,1,0..n-1))) }
function pascal2(n) { a = I(n) a[.,1] = J(n,1,1) for (i=3; i<=n; i++) { a[i,2..i-1] = a[i-1,2..i-1]+a[i-1,1..i-2] } return(a) }
function pascal3(n) { a = J(n,n,0) for (i=1; i<n; i++) { a[i+1,i] = i } s = p = I(n) k = 1 for (i=0; i<n; i++) { p = p*a/k++ s = s+p } return(s) } end</lang>
One could trivially write functions for the upper matrix (same operations with transposed matrices). The symmetric matrix can be generated using loops. However, when the lower matrix is known the other two are immediately deduced:
<lang stata>: a = pascal3(5)
- a
1 2 3 4 5 +---------------------+ 1 | 1 0 0 0 0 | 2 | 1 1 0 0 0 | 3 | 1 2 1 0 0 | 4 | 1 3 3 1 0 | 5 | 1 4 6 4 1 | +---------------------+
- a'
1 2 3 4 5 +---------------------+ 1 | 1 1 1 1 1 | 2 | 0 1 2 3 4 | 3 | 0 0 1 3 6 | 4 | 0 0 0 1 4 | 5 | 0 0 0 0 1 | +---------------------+
- a*a'
[symmetric]
1 2 3 4 5 +--------------------------+ 1 | 1 | 2 | 1 2 | 3 | 1 3 6 | 4 | 1 4 10 20 | 5 | 1 5 15 35 70 | +--------------------------+</lang>
The last is a symmetric matrix, but Stata only shows the lower triangular part of symmetric matrices.
Tcl
<lang Tcl> package require math
namespace eval pascal {
proc upper {n} {
for {set i 0} {$i < $n} {incr i} {
for {set j 0} {$j < $n} {incr j} {
puts -nonewline \t[::math::choose $j $i]
}
puts ""
}
}
proc lower {n} {
for {set i 0} {$i < $n} {incr i} {
for {set j 0} {$j < $n} {incr j} {
puts -nonewline \t[::math::choose $i $j]
}
puts ""
}
}
proc symmetric {n} {
for {set i 0} {$i < $n} {incr i} {
for {set j 0} {$j < $n} {incr j} {
puts -nonewline \t[::math::choose [expr {$i+$j}] $i]
}
puts ""
}
}
}
foreach type {upper lower symmetric} {
puts "\n* $type" pascal::$type 5
} </lang>
- Output:
* upper
1 1 1 1 1
0 1 2 3 4
0 0 1 3 6
0 0 0 1 4
0 0 0 0 1
* lower
1 0 0 0 0
1 1 0 0 0
1 2 1 0 0
1 3 3 1 0
1 4 6 4 1
* symmetric
1 1 1 1 1
1 2 3 4 5
1 3 6 10 15
1 4 10 20 35
1 5 15 35 70
VBA
<lang vb>Option Base 1 Private Function pascal_upper(n As Integer)
Dim res As Variant: ReDim res(n, n)
For j = 1 To n
res(1, j) = 1
Next j
For i = 2 To n
res(i, 1) = 0
For j = 2 To i
res(j, i) = res(j, i - 1) + res(j - 1, i - 1)
Next j
For j = i + 1 To n
res(j, i) = 0
Next j
Next i
pascal_upper = res
End Function
Private Function pascal_symmetric(n As Integer)
Dim res As Variant: ReDim res(n, n)
For i = 1 To n
res(i, 1) = 1
res(1, i) = 1
Next i
For i = 2 To n
For j = 2 To n
res(i, j) = res(i - 1, j) + res(i, j - 1)
Next j
Next i
pascal_symmetric = res
End Function
Private Sub pp(m As Variant)
For i = 1 To UBound(m)
For j = 1 To UBound(m, 2)
Debug.Print Format(m(i, j), "@@@");
Next j
Debug.Print
Next i
End Sub
Public Sub main()
Debug.Print "=== Pascal upper matrix ===" pp pascal_upper(5) Debug.Print "=== Pascal lower matrix ===" pp WorksheetFunction.Transpose(pascal_upper(5)) Debug.Print "=== Pascal symmetrical matrix ===" pp pascal_symmetric(5)End Sub</lang>
- Output:
=== Pascal upper matrix === 1 1 1 1 1 0 1 2 3 4 0 0 1 3 6 0 0 0 1 4 0 0 0 0 1 === Pascal lower matrix === 1 0 0 0 0 1 1 0 0 0 1 2 1 0 0 1 3 3 1 0 1 4 6 4 1 === Pascal symmetrical matrix === 1 1 1 1 1 1 2 3 4 5 1 3 6 10 15 1 4 10 20 35 1 5 15 35 70
VBScript
<lang vb> Function pascal_upper(i,j) WScript.StdOut.Write "Pascal Upper" WScript.StdOut.WriteLine For l = i To j For m = i To j If l <= m Then WScript.StdOut.Write binomial(m,l) & vbTab Else WScript.StdOut.Write 0 & vbTab End If Next WScript.StdOut.WriteLine Next WScript.StdOut.WriteLine End Function
Function pascal_lower(i,j) WScript.StdOut.Write "Pascal Lower" WScript.StdOut.WriteLine For l = i To j For m = i To j If l >= m Then WScript.StdOut.Write binomial(l,m) & vbTab Else WScript.StdOut.Write 0 & vbTab End If Next WScript.StdOut.WriteLine Next WScript.StdOut.WriteLine End Function
Function pascal_symmetric(i,j) WScript.StdOut.Write "Pascal Symmetric" WScript.StdOut.WriteLine For l = i To j For m = i To j WScript.StdOut.Write binomial(l+m,m) & vbTab Next WScript.StdOut.WriteLine Next End Function
Function binomial(n,k) binomial = factorial(n)/(factorial(n-k)*factorial(k)) End Function
Function factorial(n) If n = 0 Then factorial = 1 Else For i = n To 1 Step -1 If i = n Then factorial = n Else factorial = factorial * i End If Next End If End Function
'Test driving Call pascal_upper(0,4) Call pascal_lower(0,4) Call pascal_symmetric(0,4) </lang>
- Output:
Pascal Upper 1 1 1 1 1 0 1 2 3 4 0 0 1 3 6 0 0 0 1 4 0 0 0 0 1 Pascal Lower 1 0 0 0 0 1 1 0 0 0 1 2 1 0 0 1 3 3 1 0 1 4 6 4 1 Pascal Symmetric 1 1 1 1 1 1 2 3 4 5 1 3 6 10 15 1 4 10 20 35 1 5 15 35 70
Wren
<lang ecmascript>import "/fmt" for Fmt import "/math" for Int import "/matrix" for Matrix
var binomial = Fn.new { |n, k|
if (n == k) return 1
var prod = 1
var i = n - k + 1
while (i <= n) {
prod = prod * i
i = i + 1
}
return prod / Int.factorial(k)
}
var pascalUpperTriangular = Fn.new { |n|
var m = List.filled(n, null)
for (i in 0...n) {
m[i] = List.filled(n, 0)
for (j in 0...n) m[i][j] = binomial.call(j, i)
}
return Matrix.new(m)
}
var pascalSymmetric = Fn.new { |n|
var m = List.filled(n, null)
for (i in 0...n) {
m[i] = List.filled(n, 0)
for (j in 0...n) m[i][j] = binomial.call(i+j, i)
}
return Matrix.new(m)
}
var pascalLowerTriangular = Fn.new { |n| pascalSymmetric.call(n).cholesky() }
var n = 5 System.print("Pascal upper-triangular matrix:") Fmt.mprint(pascalUpperTriangular.call(n), 2, 0) System.print("\nPascal lower-triangular matrix:") Fmt.mprint(pascalLowerTriangular.call(n), 2, 0) System.print("\nPascal symmetric matrix:") Fmt.mprint(pascalSymmetric.call(n), 2, 0)</lang>
- Output:
Pascal upper-triangular matrix: | 1 1 1 1 1| | 0 1 2 3 4| | 0 0 1 3 6| | 0 0 0 1 4| | 0 0 0 0 1| Pascal lower-triangular matrix: | 1 0 0 0 0| | 1 1 0 0 0| | 1 2 1 0 0| | 1 3 3 1 0| | 1 4 6 4 1| Pascal symmetric matrix: | 1 1 1 1 1| | 1 2 3 4 5| | 1 3 6 10 15| | 1 4 10 20 35| | 1 5 15 35 70|
zkl
<lang zkl>fcn binomial(n,k){ (1).reduce(k,fcn(p,i,n){ p*(n-i+1)/i },1,n) } fcn pascal_upp(n){ [[(i,j); n; n; '{ binomial(j,i) }]]:toMatrix(_) } // [[..]] is list comprehension fcn pascal_low(n){ (i,j); n; n; binomial:toMatrix(_) } fcn pascal_sym(n){ [[(i,j); n; n; '{ binomial(i+j,i) }]]:toMatrix(_) } fcn toMatrix(ns){ // turn a string of numbers into a square matrix (list of lists)
cols:=ns.len().toFloat().sqrt().toInt(); ns.pump(List,T(Void.Read,cols-1),List.create)
}</lang> <lang zkl>fcn prettyPrint(m){ // m is a list of lists
fmt:=("%3d "*m.len() + "\n").fmt;
m.pump(String,'wrap(col){ fmt(col.xplode()) });
} const N=5; println("Upper:\n", pascal_upp(N):prettyPrint(_)); println("Lower:\n", pascal_low(N):prettyPrint(_)); println("Symmetric:\n",pascal_sym(N):prettyPrint(_));</lang>
- Output:
Upper: 1 1 1 1 1 0 1 2 3 4 0 0 1 3 6 0 0 0 1 4 0 0 0 0 1 Lower: 1 0 0 0 0 1 1 0 0 0 1 2 1 0 0 1 3 3 1 0 1 4 6 4 1 Symmetric: 1 1 1 1 1 1 2 3 4 5 1 3 6 10 15 1 4 10 20 35 1 5 15 35 70
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