Lah numbers
Lah numbers, sometimes referred to as Stirling numbers of the third kind, are coefficients of polynomial expansions expressing rising factorials in terms of falling factorials.
You are encouraged to solve this task according to the task description, using any language you may know.
Unsigned Lah numbers count the number of ways a set of n elements can be partitioned into k non-empty linearly ordered subsets.
Lah numbers are closely related to Stirling numbers of the first & second kinds, and may be derived from them.
Lah numbers obey the identities and relations:
L(n, 0), L(0, k) = 0 # for n, k > 0
L(n, n) = 1
L(n, 1) = n!
L(n, k) = ( n! * (n - 1)! ) / ( k! * (k - 1)! ) / (n - k)! # For unsigned Lah numbers
or
L(n, k) = (-1)**n * ( n! * (n - 1)! ) / ( k! * (k - 1)! ) / (n - k)! # For signed Lah numbers
- Task
-
- Write a routine (function, procedure, whatever) to find unsigned Lah numbers. There are several methods to generate unsigned Lah numbers. You are free to choose the most appropriate for your language. If your language has a built-in, or easily, publicly available library implementation, it is acceptable to use that.
- Using the routine, generate and show here, on this page, a table (or triangle) showing the unsigned Lah numbers, L(n, k), up to L(12, 12). it is optional to show the row / column for n == 0 and k == 0. It is optional to show places where L(n, k) == 0 (when k > n).
- If your language supports large integers, find and show here, on this page, the maximum value of L(n, k) where n == 100.
- See also
- Related Tasks
11l
<lang 11l>F lah(BigInt n, BigInt k)
I k == 1
R factorial(n)
I k == n
R BigInt(1)
I k > n
R BigInt(0)
I k < 1 | n < 1
R BigInt(0)
R (factorial(n) * factorial(n - 1)) I/ (factorial(k) * factorial(k - 1)) I/ factorial(n - k)
print(‘Unsigned Lah numbers: L(n, k):’) print(‘n/k ’, end' ‘ ’) L(i) 13
print(‘#11’.format(i), end' ‘ ’)
print() L(row) 13
print(‘#<4’.format(row), end' ‘ ’)
L(i) 0 .. row
V l = lah(row, i)
print(‘#11’.format(l), end' ‘ ’)
print()
print("\nMaximum value from the L(100, *) row:") V maxVal = max((0.<100).map(a -> lah(100, a))) print(maxVal)</lang>
- Output:
Unsigned Lah numbers: L(n, k): n/k 0 1 2 3 4 5 6 7 8 9 10 11 12 0 1 1 0 1 2 0 2 1 3 0 6 6 1 4 0 24 36 12 1 5 0 120 240 120 20 1 6 0 720 1800 1200 300 30 1 7 0 5040 15120 12600 4200 630 42 1 8 0 40320 141120 141120 58800 11760 1176 56 1 9 0 362880 1451520 1693440 846720 211680 28224 2016 72 1 10 0 3628800 16329600 21772800 12700800 3810240 635040 60480 3240 90 1 11 0 39916800 199584000 299376000 199584000 69854400 13970880 1663200 118800 4950 110 1 12 0 479001600 2634508800 4390848000 3293136000 1317254400 307359360 43908480 3920400 217800 7260 132 1 Maximum value from the L(100, *) row: 44519005448993144810881324947684737529186447692709328597242209638906324913313742508392928375354932241404408343800007105650554669129521241784320000000000000000000000
ALGOL 68
Uses Algol 68G's LONG LONG INT which has programmer-specifiable precision, so can find Lah numbers with n = 100 and need not use the scaling "trick" used by the Algol W sample. <lang algol68>BEGIN # calculate Lah numbers upto L( 12, 12 ) #
PR precision 400 PR # set the precision for LONG LONG INT #
# returns Lah Number L( n, k ), f must be a table of factorials to at least n #
PROC lah = ( INT n, k, []LONG LONG INT f )LONG LONG INT:
IF n = k THEN 1
ELIF n = 0 OR k = 0 THEN 0
ELIF k = 1 THEN f[ n ]
ELIF k > n THEN 0
ELSE
# general case: ( n! * ( n - 1 )! ) / ( k! * ( k - 1 )! ) / ( n - k )! #
# we re-arrange the above to: #
# ( n! / k! ) -- t1 #
# * ( ( n - 1 )! / ( k - 1 )! ) -- t2 #
# / ( n - k )! #
LONG LONG INT t1 = f[ n ] OVER f[ k ];
LONG LONG INT t2 = f[ n - 1 ] OVER f[ k - 1 ];
( t1 * t2 ) OVER f[ n - k ]
FI # lah # ;
INT max n = 100; # max n for Lah Numbers #
INT max display = 12; # max n to display L( n, k ) values #
# table of factorials up to max n #
[ 1 : max n ]LONG LONG INT factorial;
BEGIN
LONG LONG INT f := 1;
FOR i TO UPB factorial DO factorial[ i ] := f *:= i OD
END;
# show the Lah numbers #
print( ( "Unsigned Lah numbers", newline ) );
print( ( "n/k 0" ) );
FOR i FROM 1 TO max display DO print( ( whole( i, -11 ) ) ) OD;
print( ( newline ) );
FOR n FROM 0 TO max display DO
print( ( whole( n, -2 ) ) );
print( ( whole( lah( n, 0, factorial ), -4 ) ) );
FOR k FROM 1 TO n DO
print( ( whole( lah( n, k, factorial ), -11 ) ) )
OD;
print( ( newline ) )
OD;
# maximum value of a Lah number for n = 100 #
LONG LONG INT max 100 := 0;
FOR k FROM 0 TO 100 DO
LONG LONG INT lah n k = lah( 100, k, factorial );
IF lah n k > max 100 THEN max 100 := lah n k FI
OD;
print( ( "maximum Lah number for n = 100: ", whole( max 100, 0 ), newline ) )
END</lang>
- Output:
Unsigned Lah numbers n/k 0 1 2 3 4 5 6 7 8 9 10 11 12 0 1 1 0 1 2 0 2 1 3 0 6 6 1 4 0 24 36 12 1 5 0 120 240 120 20 1 6 0 720 1800 1200 300 30 1 7 0 5040 15120 12600 4200 630 42 1 8 0 40320 141120 141120 58800 11760 1176 56 1 9 0 362880 1451520 1693440 846720 211680 28224 2016 72 1 10 0 3628800 16329600 21772800 12700800 3810240 635040 60480 3240 90 1 11 0 39916800 199584000 299376000 199584000 69854400 13970880 1663200 118800 4950 110 1 12 0 479001600 2634508800 4390848000 3293136000 1317254400 307359360 43908480 3920400 217800 7260 132 1 maximum Lah number for n = 100: 44519005448993144810881324947684737529186447692709328597242209638906324913313742508392928375354932241404408343800007105650554669129521241784320000000000000000000000
ALGOL W
Algol W's only integer type is signed 32 bit integers.
As L(12,2), L(12,3) and L(12,4) are too large for signed 32 bit integers, this sample scales the result by an appropriate power of 10 to enable the table to be printed up to L(12,12). Luckily, the problematic L(12,k) values all have at least 2 trailing zeros.
<lang algolw>begin % calculate Lah numbers upto L( 12, 12 ) %
% sets lahNumber to L( n, k ), lahScale is returned as the power of 10 %
% lahNumber should be multiplied by %
% f must be a table of factorials to at least n %
procedure L ( integer value n, k
; integer array f ( * )
; integer result lahNumber, lahScale
) ;
if n = k then begin lahNumber := 1; lahScale := 0 end
else if n = 0 or k = 0 then begin lahNumber := 0; lahScale := 0 end
else if k = 1 then begin lahNumber := f( n ); lahScale := 0 end
else if k > n then begin lahNumber := 0; lahScale := 0 end
else begin
% general case: ( n! * ( n - 1 )! ) / ( k! * ( k - 1 )! ) / ( n - k )! %
% Algol W has only 32 bit signed integers so we need to avoid overflow %
% we re-arrange the above to: %
% ( n! / k! / ( n - k ) ! ) -- t1 %
% * ( ( n - 1 )! / ( k - 1 )! -- t2 %
% and if necessary, reduce t1 and t2 by powers of 10 %
integer t1, t2, d1, d2, v;
t1 := f( n ) div f( k ) div f( n - k );
t2 := f( n - 1 ) div f( k - 1 );
% calculate the number of digits for t1 and t2 %
lahScale := d1 := d2 := 0;
v := t1; while v > 0 do begin d1 := d1 + 1; v := v div 10 end;
v := t2; while v > 0 do begin d2 := d2 + 1; v := v div 10 end;
if d1 + d2 > 8 then begin
% the result will overflow reduce t1 and t2 by an appropriate power %
% of 10 and set lahScale accordingly %
while t1 rem 10 = 0 do begin lahScale := lahScale + 1; t1 := t1 div 10 end;
while t2 rem 10 = 0 do begin lahScale := lahScale + 1; t2 := t2 div 10 end;
end if_d1_plus_d2_gt_8;
lahNumber := t1 * t2
end L;
% table of factorials up to 12 %
integer array factorial ( 1 :: 12 );
% compute the factorials %
begin
integer f; f := 1;
for i := 1 until 12 do begin f := f * i; factorial( i ) := f end
end;
% show the Lah numbers %
write( "Unsigned Lah numbers" );
write( "n/k " );
for i := 0 until 12 do writeon( i_w := 11, s_w := 0, i );
for n := 0 until 12 do begin
write( s_w := 2, i_w := 2, n );
for k := 0 until n do begin
integer lahNumber, lahScale;
L( n, k, factorial, lahNumber, lahScale );
writeon( S_W := 0, i_w := 11 - lahScale, lahNumber );
for s := 1 until lahScale do writeon( s_w := 0, "0" )
end for_k
end for_n
end.</lang>
- Output:
Unsigned Lah numbers n/k 0 1 2 3 4 5 6 7 8 9 10 11 12 0 1 1 0 1 2 0 2 1 3 0 6 6 1 4 0 24 36 12 1 5 0 120 240 120 20 1 6 0 720 1800 1200 300 30 1 7 0 5040 15120 12600 4200 630 42 1 8 0 40320 141120 141120 58800 11760 1176 56 1 9 0 362880 1451520 1693440 846720 211680 28224 2016 72 1 10 0 3628800 16329600 21772800 12700800 3810240 635040 60480 3240 90 1 11 0 39916800 199584000 299376000 199584000 69854400 13970880 1663200 118800 4950 110 1 12 0 479001600 2634508800 4390848000 3293136000 1317254400 307359360 43908480 3920400 217800 7260 132 1
Arturo
<lang rebol>factorial: function [n]-> product 1..n
lah: function [n,k][
if k=1 -> return factorial n if k=n -> return 1 if k>n -> return 0 if or? k<1 n<1 -> return 0 return (((factorial n)*factorial n-1) / ((factorial k) * factorial k-1)) / factorial n-k
]
print @["n/k"] ++ map to [:string] 1..12 's -> pad s 10 print repeat "-" 136 loop 1..12 'x [
print @[pad to :string x 3] ++ map to [:string] map 1..12 'y -> lah x y 's -> pad s 10
]</lang>
- Output:
n/k 1 2 3 4 5 6 7 8 9 10 11 12 ---------------------------------------------------------------------------------------------------------------------------------------- 1 1 0 0 0 0 0 0 0 0 0 0 0 2 2 1 0 0 0 0 0 0 0 0 0 0 3 6 6 1 0 0 0 0 0 0 0 0 0 4 24 36 12 1 0 0 0 0 0 0 0 0 5 120 240 120 20 1 0 0 0 0 0 0 0 6 720 1800 1200 300 30 1 0 0 0 0 0 0 7 5040 15120 12600 4200 630 42 1 0 0 0 0 0 8 40320 141120 141120 58800 11760 1176 56 1 0 0 0 0 9 362880 1451520 1693440 846720 211680 28224 2016 72 1 0 0 0 10 3628800 16329600 21772800 12700800 3810240 635040 60480 3240 90 1 0 0 11 39916800 199584000 299376000 199584000 69854400 13970880 1663200 118800 4950 110 1 0 12 479001600 2634508800 4390848000 3293136000 1317254400 307359360 43908480 3920400 217800 7260 132 1
AWK
<lang AWK>
- syntax: GAWK -f LAH_NUMBERS.AWK
- converted from C
BEGIN {
print("unsigned Lah numbers: L(n,k)")
printf("n/k")
for (i=0; i<13; i++) {
printf("%11d",i)
}
printf("\n")
for (row=0; row<13; row++) {
printf("%-3d",row)
for (i=0; i<row+1; i++) {
printf(" %10d",lah(row,i))
}
printf("\n")
}
exit(0)
} function factorial(n, res) {
res = 1
if (n == 0) { return(res) }
while (n > 0) { res *= n-- }
return(res)
} function lah(n,k) {
if (k == 1) { return factorial(n) }
if (k == n) { return(1) }
if (k > n) { return(0) }
if (k < 1 || n < 1) { return(0) }
return (factorial(n) * factorial(n-1)) / (factorial(k) * factorial(k-1)) / factorial(n-k)
} </lang>
- Output:
unsigned Lah numbers: L(n,k) n/k 0 1 2 3 4 5 6 7 8 9 10 11 12 0 1 1 0 1 2 0 2 1 3 0 6 6 1 4 0 24 36 12 1 5 0 120 240 120 20 1 6 0 720 1800 1200 300 30 1 7 0 5040 15120 12600 4200 630 42 1 8 0 40320 141120 141120 58800 11760 1176 56 1 9 0 362880 1451520 1693440 846720 211680 28224 2016 72 1 10 0 3628800 16329600 21772800 12700800 3810240 635040 60480 3240 90 1 11 0 39916800 199584000 299376000 199584000 69854400 13970880 1663200 118800 4950 110 1 12 0 479001600 2634508800 4390848000 3293136000 1317254400 307359360 43908480 3920400 217800 7260 132 1
BQN
F is a function that computes the factorial of a number. It returns 0 for negative numbers.
Lah computes the lah function, called as n Lah k.
The last line builds a table where the rows represent n and columns represent k.
<lang bqn>F ← (≥⟜0)◶⟨0,×´1+↕⟩
Lah ← {
𝕨 𝕊 0: 0; 0 𝕊 𝕩: 0; 𝕨 𝕊 1: F 𝕨; n 𝕊 k: (n=k)⊑⟨((n ×○F n-1) ÷ k ×○F k-1) (0=⊢)◶÷‿0 F n-k, 1⟩
}
•Show Lah⌜˜↕12</lang><lang>┌─ ╵ 0 0 0 0 0 0 0 0 0 0 0 0
0 1 0 0 0 0 0 0 0 0 0 0
0 2 1 0 0 0 0 0 0 0 0 0
0 6 6 1 0 0 0 0 0 0 0 0
0 24 36 12 1 0 0 0 0 0 0 0
0 120 240 120 20 1 0 0 0 0 0 0
0 720 1800 1200 300 30 1 0 0 0 0 0
0 5040 15120 12600 4200 630 42 1 0 0 0 0
0 40320 141120 141120 58800 11760 1176 56 1 0 0 0
0 362880 1451520 1693440 846720 211680 28224 2016 72 1 0 0
0 3628800 16329600 21772800 12700800 3810240 635040 60480 3240 90 1 0
0 39916800 199584000 299376000 199584000 69854400 13970880 1663200 118800 4950 110 1
┘</lang>
C
<lang c>#include <stdint.h>
- include <stdio.h>
uint64_t factorial(uint64_t n) {
uint64_t res = 1; if (n == 0) return res; while (n > 0) res *= n--; return res;
}
uint64_t lah(uint64_t n, uint64_t k) {
if (k == 1) return factorial(n); if (k == n) return 1; if (k > n) return 0; if (k < 1 || n < 1) return 0; return (factorial(n) * factorial(n - 1)) / (factorial(k) * factorial(k - 1)) / factorial(n - k);
}
int main() {
int row, i;
printf("Unsigned Lah numbers: L(n, k):\n");
printf("n/k ");
for (i = 0; i < 13; i++) {
printf("%10d ", i);
}
printf("\n");
for (row = 0; row < 13; row++) {
printf("%-3d", row);
for (i = 0; i < row + 1; i++) {
uint64_t l = lah(row, i);
printf("%11lld", l);
}
printf("\n");
}
return 0;
}</lang>
- Output:
Unsigned Lah numbers: L(n, k): n/k 0 1 2 3 4 5 6 7 8 9 10 11 12 0 1 1 0 1 2 0 2 1 3 0 6 6 1 4 0 24 36 12 1 5 0 120 240 120 20 1 6 0 720 1800 1200 300 30 1 7 0 5040 15120 12600 4200 630 42 1 8 0 40320 141120 141120 58800 11760 1176 56 1 9 0 362880 1451520 1693440 846720 211680 28224 2016 72 1 10 0 3628800 16329600 21772800 12700800 3810240 635040 60480 3240 90 1 11 0 39916800 199584000 299376000 199584000 69854400 13970880 1663200 118800 4950 110 1 12 0 479001600 2634508800 4390848000 3293136000 1317254400 307359360 43908480 3920400 217800 7260 132 1
C#
<lang csharp>using System; using System.Linq; using System.Numerics;
namespace LahNumbers {
class Program {
static BigInteger Factorial(BigInteger n) {
if (n == 0) return 1;
BigInteger res = 1;
while (n > 0) {
res *= n--;
}
return res;
}
static BigInteger Lah(BigInteger n, BigInteger k) {
if (k == 1) return Factorial(n);
if (k == n) return 1;
if (k > n) return 0;
if (k < 1 || n < 1) return 0;
return (Factorial(n) * Factorial(n - 1)) / (Factorial(k) * Factorial(k - 1)) / Factorial(n - k);
}
static void Main() {
Console.WriteLine("Unsigned Lah numbers: L(n, k):");
Console.Write("n/k ");
foreach (var i in Enumerable.Range(0, 13)) {
Console.Write("{0,10} ", i);
}
Console.WriteLine();
foreach (var row in Enumerable.Range(0, 13)) {
Console.Write("{0,-3}", row);
foreach (var i in Enumerable.Range(0, row + 1)) {
var l = Lah(row, i);
Console.Write("{0,11}", l);
}
Console.WriteLine();
}
Console.WriteLine("\nMaximum value from the L(100, *) row:");
var maxVal = Enumerable.Range(0, 100).Select(a => Lah(100, a)).Max();
Console.WriteLine(maxVal);
}
}
}</lang>
- Output:
0 1 1 0 1 2 0 2 1 3 0 6 6 1 4 0 24 36 12 1 5 0 120 240 120 20 1 6 0 720 1800 1200 300 30 1 7 0 5040 15120 12600 4200 630 42 1 8 0 40320 141120 141120 58800 11760 1176 56 1 9 0 362880 1451520 1693440 846720 211680 28224 2016 72 1 10 0 3628800 16329600 21772800 12700800 3810240 635040 60480 3240 90 1 11 0 39916800 199584000 299376000 199584000 69854400 13970880 1663200 118800 4950 110 1 12 0 479001600 2634508800 4390848000 3293136000 1317254400 307359360 43908480 3920400 217800 7260 132 1 Maximum value from the L(100, *) row: 44519005448993144810881324947684737529186447692709328597242209638906324913313742508392928375354932241404408343800007105650554669129521241784320000000000000000000000
C++
<lang cpp>// Reference: https://en.wikipedia.org/wiki/Lah_number#Identities_and_relations
- include <algorithm>
- include <iomanip>
- include <iostream>
- include <map>
- include <gmpxx.h>
using integer = mpz_class;
class unsigned_lah_numbers { public:
integer get(int n, int k);
private:
std::map<std::pair<int, int>, integer> cache_;
};
integer unsigned_lah_numbers::get(int n, int k) {
if (k == n)
return 1;
if (k == 0 || k > n)
return 0;
auto p = std::make_pair(n, k);
auto i = cache_.find(p);
if (i != cache_.end())
return i->second;
integer result = (n - 1 + k) * get(n - 1, k) + get(n - 1, k - 1);
cache_.emplace(p, result);
return result;
}
void print_lah_numbers(unsigned_lah_numbers& uln, int n) {
std::cout << "Unsigned Lah numbers up to L(12,12):\nn/k";
for (int j = 1; j <= n; ++j)
std::cout << std::setw(11) << j;
std::cout << '\n';
for (int i = 1; i <= n; ++i) {
std::cout << std::setw(2) << i << ' ';
for (int j = 1; j <= i; ++j)
std::cout << std::setw(11) << uln.get(i, j);
std::cout << '\n';
}
}
int main() {
unsigned_lah_numbers uln;
print_lah_numbers(uln, 12);
std::cout << "Maximum value of L(n,k) where n == 100:\n";
integer max = 0;
for (int k = 0; k <= 100; ++k)
max = std::max(max, uln.get(100, k));
std::cout << max << '\n';
return 0;
}</lang>
- Output:
Unsigned Lah numbers up to L(12,12): n/k 1 2 3 4 5 6 7 8 9 10 11 12 1 1 2 2 1 3 6 6 1 4 24 36 12 1 5 120 240 120 20 1 6 720 1800 1200 300 30 1 7 5040 15120 12600 4200 630 42 1 8 40320 141120 141120 58800 11760 1176 56 1 9 362880 1451520 1693440 846720 211680 28224 2016 72 1 10 3628800 16329600 21772800 12700800 3810240 635040 60480 3240 90 1 11 39916800 199584000 299376000 199584000 69854400 13970880 1663200 118800 4950 110 1 12 479001600 2634508800 4390848000 3293136000 1317254400 307359360 43908480 3920400 217800 7260 132 1 Maximum value of L(n,k) where n == 100: 44519005448993144810881324947684737529186447692709328597242209638906324913313742508392928375354932241404408343800007105650554669129521241784320000000000000000000000
D
<lang d>import std.algorithm : map; import std.bigint; import std.range; import std.stdio;
BigInt factorial(BigInt n) {
if (n == 0) return BigInt(1);
BigInt res = 1;
while (n > 0) {
res *= n--;
}
return res;
}
BigInt lah(BigInt n, BigInt k) {
if (k == 1) return factorial(n); if (k == n) return BigInt(1); if (k > n) return BigInt(0); if (k < 1 || n < 1) return BigInt(0); return (factorial(n) * factorial(n - 1)) / (factorial(k) * factorial(k - 1)) / factorial(n - k);
}
auto max(R)(R r) if (isInputRange!R) {
alias T = ElementType!R; T v = T.init;
while (!r.empty) {
if (v < r.front) {
v = r.front;
}
r.popFront;
}
return v;
}
void main() {
writeln("Unsigned Lah numbers: L(n, k):");
write("n/k ");
foreach (i; 0..13) {
writef("%10d ", i);
}
writeln();
foreach (row; 0..13) {
writef("%-3d", row);
foreach (i; 0..row+1) {
auto l = lah(BigInt(row), BigInt(i));
writef("%11d", l);
}
writeln();
}
writeln("\nMaximum value from the L(100, *) row:");
auto lambda = (int a) => lah(BigInt(100), BigInt(a));
writeln(iota(0, 100).map!lambda.max);
}</lang>
- Output:
Unsigned Lah numbers: L(n, k): n/k 0 1 2 3 4 5 6 7 8 9 10 11 12 0 1 1 0 1 2 0 2 1 3 0 6 6 1 4 0 24 36 12 1 5 0 120 240 120 20 1 6 0 720 1800 1200 300 30 1 7 0 5040 15120 12600 4200 630 42 1 8 0 40320 141120 141120 58800 11760 1176 56 1 9 0 362880 1451520 1693440 846720 211680 28224 2016 72 1 10 0 3628800 16329600 21772800 12700800 3810240 635040 60480 3240 90 1 11 0 39916800 199584000 299376000 199584000 69854400 13970880 1663200 118800 4950 110 1 12 0 479001600 2634508800 4390848000 3293136000 1317254400 307359360 43908480 3920400 217800 7260 132 1 Maximum value from the L(100, *) row: 44519005448993144810881324947684737529186447692709328597242209638906324913313742508392928375354932241404408343800007105650554669129521241784320000000000000000000000
Factor
<lang factor>USING: combinators combinators.short-circuit formatting infix io kernel locals math math.factorials math.ranges prettyprint sequences ; IN: rosetta-code.lah-numbers
! Yes, Factor can do infix arithmetic with local variables! ! This is a good use case for it.
INFIX:: (lah) ( n k -- m )
( factorial(n) * factorial(n-1) ) / ( factorial(k) * factorial(k-1) ) / factorial(n-k) ;
- lah ( n k -- m )
{
{ [ k 1 = ] [ n factorial ] }
{ [ k n = ] [ 1 ] }
{ [ k n > ] [ 0 ] }
{ [ k 1 < n 1 < or ] [ 0 ] }
[ n k (lah) ]
} cond ;
"Unsigned Lah numbers: n k lah:" print "n\\k" write 13 dup [ "%11d" printf ] each-integer nl
<iota> [
dup dup "%-2d " printf [0,b] [
lah "%11d" printf
] with each nl
] each nl
"Maximum value from the 100 _ lah row:" print 100 [0,b] [ 100 swap lah ] map supremum .</lang>
- Output:
Unsigned Lah numbers: n k lah: n\k 0 1 2 3 4 5 6 7 8 9 10 11 12 0 1 1 0 1 2 0 2 1 3 0 6 6 1 4 0 24 36 12 1 5 0 120 240 120 20 1 6 0 720 1800 1200 300 30 1 7 0 5040 15120 12600 4200 630 42 1 8 0 40320 141120 141120 58800 11760 1176 56 1 9 0 362880 1451520 1693440 846720 211680 28224 2016 72 1 10 0 3628800 16329600 21772800 12700800 3810240 635040 60480 3240 90 1 11 0 39916800 199584000 299376000 199584000 69854400 13970880 1663200 118800 4950 110 1 12 0 479001600 2634508800 4390848000 3293136000 1317254400 307359360 43908480 3920400 217800 7260 132 1 Maximum value from the 100 _ lah row: 44519005448993144810881324947684737529186447692709328597242209638906324913313742508392928375354932241404408343800007105650554669129521241784320000000000000000000000
FreeBASIC
<lang FreeBASIC>function factorial( n as uinteger ) as ulongint
if n = 0 then return 1 return n*factorial(n-1)
end function
function s_Lah( n as uinteger, k as uinteger ) as ulongint
if n = k then return 1 if n = 0 orelse k = 0 then return 0 if k = 1 then return factorial(n) return ((-1)^n)*(factorial(n)*factorial(n - 1))/(factorial(k)*factorial(k - 1))/factorial(n - k)
end function
function u_Lah( n as uinteger, k as uinteger ) as ulongint
return abs(s_Lah(n,k))
end function
function padto( i as ubyte, j as integer ) as string
return wspace(i-len(str(j)))+str(j)
end function
print "Unsiged Lah numbers" print dim as string outstr = " k" for k as integer =0 to 12
outstr += padto(12, k)
next k print outstr print " n" for n as integer = 0 to 12
outstr = padto(2, n)+" "
for k as integer = 0 to n
outstr += padto(12, u_Lah(n, k))
next k
print outstr
next n</lang>
- Output:
Unsiged Lah numbers k 0 1 2 3 4 5 6 7 8 9 10 11 12 n 0 1 1 0 1 2 0 2 1 3 0 6 6 1 4 0 24 36 12 1 5 0 120 240 120 20 1 6 0 720 1800 1200 300 30 1 7 0 5040 15120 12600 4200 630 42 1 8 0 40320 141120 141120 58800 11760 1176 56 1 9 0 362880 1451520 1693440 846720 211680 28224 2016 72 1 10 0 3628800 16329600 21772800 12700800 3810240 635040 60480 3240 90 1 11 0 39916800 199584000 299376000 199584000 69854400 13970880 1663200 118800 4950 110 1 12 0 479001600 2634508800 4390848000 3293136000 1317254400 307359360 43908480 3920400 217800 7260 132 1
Fōrmulæ
Fōrmulæ programs are not textual, visualization/edition of programs is done showing/manipulating structures but not text. Moreover, there can be multiple visual representations of the same program. Even though it is possible to have textual representation —i.e. XML, JSON— they are intended for storage and transfer purposes more than visualization and edition.
Programs in Fōrmulæ are created/edited online in its website, However they run on execution servers. By default remote servers are used, but they are limited in memory and processing power, since they are intended for demonstration and casual use. A local server can be downloaded and installed, it has no limitations (it runs in your own computer). Because of that, example programs can be fully visualized and edited, but some of them will not run if they require a moderate or heavy computation/memory resources, and no local server is being used.
In this page you can see the program(s) related to this task and their results.
Go
<lang go>package main
import (
"fmt" "math/big"
)
func main() {
limit := 100
last := 12
unsigned := true
l := make([][]*big.Int, limit+1)
for n := 0; n <= limit; n++ {
l[n] = make([]*big.Int, limit+1)
for k := 0; k <= limit; k++ {
l[n][k] = new(big.Int)
}
l[n][n].SetInt64(int64(1))
if n != 1 {
l[n][1].MulRange(int64(2), int64(n))
}
}
var t big.Int
for n := 1; n <= limit; n++ {
for k := 1; k <= n; k++ {
t.Mul(l[n][1], l[n-1][1])
t.Quo(&t, l[k][1])
t.Quo(&t, l[k-1][1])
t.Quo(&t, l[n-k][1])
l[n][k].Set(&t)
if !unsigned && (n%2 == 1) {
l[n][k].Neg(l[n][k])
}
}
}
fmt.Println("Unsigned Lah numbers: l(n, k):")
fmt.Printf("n/k")
for i := 0; i <= last; i++ {
fmt.Printf("%10d ", i)
}
fmt.Printf("\n--")
for i := 0; i <= last; i++ {
fmt.Printf("-----------")
}
fmt.Println()
for n := 0; n <= last; n++ {
fmt.Printf("%2d ", n)
for k := 0; k <= n; k++ {
fmt.Printf("%10d ", l[n][k])
}
fmt.Println()
}
fmt.Println("\nMaximum value from the l(100, *) row:")
max := new(big.Int).Set(l[limit][0])
for k := 1; k <= limit; k++ {
if l[limit][k].Cmp(max) > 0 {
max.Set(l[limit][k])
}
}
fmt.Println(max)
fmt.Printf("which has %d digits.\n", len(max.String()))
}</lang>
- Output:
Unsigned Lah numbers: l(n, k): n/k 0 1 2 3 4 5 6 7 8 9 10 11 12 ------------------------------------------------------------------------------------------------------------------------------------------------- 0 1 1 0 1 2 0 2 1 3 0 6 6 1 4 0 24 36 12 1 5 0 120 240 120 20 1 6 0 720 1800 1200 300 30 1 7 0 5040 15120 12600 4200 630 42 1 8 0 40320 141120 141120 58800 11760 1176 56 1 9 0 362880 1451520 1693440 846720 211680 28224 2016 72 1 10 0 3628800 16329600 21772800 12700800 3810240 635040 60480 3240 90 1 11 0 39916800 199584000 299376000 199584000 69854400 13970880 1663200 118800 4950 110 1 12 0 479001600 2634508800 4390848000 3293136000 1317254400 307359360 43908480 3920400 217800 7260 132 1 Maximum value from the l(100, *) row: 44519005448993144810881324947684737529186447692709328597242209638906324913313742508392928375354932241404408343800007105650554669129521241784320000000000000000000000 which has 164 digits.
Haskell
<lang haskell>import Text.Printf (printf) import Control.Monad (when)
factorial :: Integral n => n -> n factorial 0 = 1 factorial n = product [1..n]
lah :: Integral n => n -> n -> n lah n k
| k == 1 = factorial n
| k == n = 1
| k > n = 0
| k < 1 || n < 1 = 0
| otherwise = f n `div` f k `div` factorial (n - k)
where
f = (*) =<< (^ 2) . factorial . pred
printLah :: (Word, Word) -> IO () printLah (n, k) = do
when (k == 0) (printf "\n%3d" n) printf "%11d" (lah n k)
main :: IO () main = do
printf "Unsigned Lah numbers: L(n, k):\nn/k" mapM_ (printf "%11d") zeroToTwelve mapM_ printLah $ (,) <$> zeroToTwelve <*> zeroToTwelve printf "\nMaximum value from the L(100, *) row:\n%d\n" (maximum $ lah 100 <$> ([0..100] :: [Integer])) where zeroToTwelve = [0..12]</lang>
- Output:
Unsigned Lah numbers: L(n, k): n/k 0 1 2 3 4 5 6 7 8 9 10 11 12 0 1 1 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 2 0 2 1 0 0 0 0 0 0 0 0 0 0 3 0 6 6 1 0 0 0 0 0 0 0 0 0 4 0 24 36 12 1 0 0 0 0 0 0 0 0 5 0 120 240 120 20 1 0 0 0 0 0 0 0 6 0 720 1800 1200 300 30 1 0 0 0 0 0 0 7 0 5040 15120 12600 4200 630 42 1 0 0 0 0 0 8 0 40320 141120 141120 58800 11760 1176 56 1 0 0 0 0 9 0 362880 1451520 1693440 846720 211680 28224 2016 72 1 0 0 0 10 0 3628800 16329600 21772800 12700800 3810240 635040 60480 3240 90 1 0 0 11 0 39916800 199584000 299376000 199584000 69854400 13970880 1663200 118800 4950 110 1 0 12 0 479001600 2634508800 4390848000 3293136000 1317254400 307359360 43908480 3920400 217800 7260 132 1 Maximum value from the L(100, *) row: 44519005448993144810881324947684737529186447692709328597242209638906324913313742508392928375354932241404408343800007105650554669129521241784320000000000000000000000
J
<lang J> NB. use: k lah n lah=: ! :(!&<: * %&!~)&x: NB. `%~' is shorter than `*inv'
NB. wory_lah translates lah to algebraic English. Monad =: :[: NB. permit only a y argument Dyad =: [: : NB. require x and y arguments but_1st =: & decrement =: <: Monad
NB. ! means either factorial or combinations (just as - means negate or subtract) factorial =: ! Monad combinations =: ! Dyad
into =: *inv Dyad times =: * Dyad extend_precision =: x: Monad wordy_lah =: ((combinations but_1st decrement) times (into but_1st factorial))but_1st extend_precision Dyad </lang>
lah&>~table~>:i.12 +------+---------------------------------------------------------------------------------------------------+ |lah&>~| 1 2 3 4 5 6 7 8 9 10 11 12| +------+---------------------------------------------------------------------------------------------------+ | 1 | 1 0 0 0 0 0 0 0 0 0 0 0| | 2 | 2 1 0 0 0 0 0 0 0 0 0 0| | 3 | 6 6 1 0 0 0 0 0 0 0 0 0| | 4 | 24 36 12 1 0 0 0 0 0 0 0 0| | 5 | 120 240 120 20 1 0 0 0 0 0 0 0| | 6 | 720 1800 1200 300 30 1 0 0 0 0 0 0| | 7 | 5040 15120 12600 4200 630 42 1 0 0 0 0 0| | 8 | 40320 141120 141120 58800 11760 1176 56 1 0 0 0 0| | 9 | 362880 1451520 1693440 846720 211680 28224 2016 72 1 0 0 0| |10 | 3628800 16329600 21772800 12700800 3810240 635040 60480 3240 90 1 0 0| |11 | 39916800 199584000 299376000 199584000 69854400 13970880 1663200 118800 4950 110 1 0| |12 |479001600 2634508800 4390848000 3293136000 1317254400 307359360 43908480 3920400 217800 7260 132 1| +------+---------------------------------------------------------------------------------------------------+ wordy_lah f. [: :(([: :!&(<: :[:) [: :* [: :(*^:_1)&(! :[:))&(x: :[:)) lah NB. 1 or 2 arguments are clear from the context ! :(!&<: * %&!~)&x: (lah/ -: wordy_lah/)~>:i.12 NB. the lah and wordy_lah tables agree 1 NB. maximum Lah value with n = 100 >./(lah~ >:@:i.)100 44519005448993144810881324947684737529186447692709328597242209638906324913313742508392928375354932241404408343800007105650554669129521241784320000000000000000000000
Java
<lang java> import java.math.BigInteger; import java.util.HashMap; import java.util.Map;
public class LahNumbers {
public static void main(String[] args) {
System.out.println("Show the unsigned Lah numbers up to n = 12:");
for ( int n = 0 ; n <= 12 ; n++ ) {
System.out.printf("%5s", n);
for ( int k = 0 ; k <= n ; k++ ) {
System.out.printf("%12s", lahNumber(n, k));
}
System.out.printf("%n");
}
System.out.println("Show the maximum value of L(100, k):");
int n = 100;
BigInteger max = BigInteger.ZERO;
for ( int k = 0 ; k <= n ; k++ ) {
max = max.max(lahNumber(n, k));
}
System.out.printf("%s", max);
}
private static Map<String,BigInteger> CACHE = new HashMap<>();
private static BigInteger lahNumber(int n, int k) {
String key = n + "," + k;
if ( CACHE.containsKey(key) ) {
return CACHE.get(key);
}
// L(n,0) = 0;
BigInteger result;
if ( n == 0 && k == 0 ) {
result = BigInteger.ONE;
}
else if ( k == 0 ) {
result = BigInteger.ZERO;
}
else if ( k > n ) {
result = BigInteger.ZERO;
}
else if ( n == 1 && k == 1 ) {
result = BigInteger.ONE;
}
else {
result = BigInteger.valueOf(n-1+k).multiply(lahNumber(n-1,k)).add(lahNumber(n-1,k-1));
}
CACHE.put(key, result);
return result;
}
} </lang>
- Output:
Show the unsigned Lah numbers up to n = 12:
0 1
1 0 1
2 0 2 1
3 0 6 6 1
4 0 24 36 12 1
5 0 120 240 120 20 1
6 0 720 1800 1200 300 30 1
7 0 5040 15120 12600 4200 630 42 1
8 0 40320 141120 141120 58800 11760 1176 56 1
9 0 362880 1451520 1693440 846720 211680 28224 2016 72 1
10 0 3628800 16329600 21772800 12700800 3810240 635040 60480 3240 90 1
11 0 39916800 199584000 299376000 199584000 69854400 13970880 1663200 118800 4950 110 1
12 0 479001600 2634508800 4390848000 3293136000 1317254400 307359360 43908480 3920400 217800 7260 132 1
Show the maximum value of L(100, k):
44519005448993144810881324947684737529186447692709328597242209638906324913313742508392928375354932241404408343800007105650554669129521241784320000000000000000000000
jq
The table can be produced by either jq or gojq (the Go implementation of jq), but gojq is required to produce the precise maximum value.
<lang jq>## Generic functions
def factorial: reduce range(2;.+1) as $i (1; . * $i);
- nCk assuming n >= k
def binomial($n; $k):
if $k > $n / 2 then binomial($n; $n-$k) else (reduce range($k+1; $n+1) as $i (1; . * $i)) as $numerator | reduce range(1;1+$n-$k) as $i ($numerator; . / $i) end;
def lpad($len): tostring | ($len - length) as $l | (" " * $l)[:$l] + .;
def max(s): reduce s as $x (-infinite; if $x > . then $x else . end);
def lah($n; $k; $signed):
if $n == $k then 1
elif $n == 0 or $k == 0 or $k > $n then 0
elif $k == 1 then $n|factorial
else
(binomial($n; $k) * binomial($n - 1; $k - 1) * (($n - $k)|factorial)) as $unsignedvalue
| if $signed and ($n % 1 == 1)
then -$unsignedvalue
else $unsignedvalue
end
end;
def lah($n; $k): lah($n;$k;false); </lang> The task <lang jq>def printlahtable($kmax):
def pad: lpad(12);
reduce range(0;$kmax+1) as $k ("n/k"|lpad(4); . + ($k|pad)),
(range(0; $kmax+1) as $n
| reduce range(0;$n+1) as $k ($n|lpad(4);
. + (lah($n; $k) | pad)) ) ;
def task:
"Unsigned Lah numbers up to n==12", printlahtable(12), "", "The maxiumum of lah(100, _) is: \(max( lah(100; range(0;101)) ))" ;
task </lang>
- Output:
Unsigned Lah numbers up to n==12 n/k 0 1 2 3 4 5 6 7 8 9 10 11 12 0 1 1 0 1 2 0 2 1 3 0 6 6 1 4 0 24 36 12 1 5 0 120 240 120 20 1 6 0 720 1800 1200 300 30 1 7 0 5040 15120 12600 4200 630 42 1 8 0 40320 141120 141120 58800 11760 1176 56 1 9 0 362880 1451520 1693440 846720 211680 28224 2016 72 1 10 0 3628800 16329600 21772800 12700800 3810240 635040 60480 3240 90 1 11 0 39916800 199584000 299376000 199584000 69854400 13970880 1663200 118800 4950 110 1 12 0 479001600 2634508800 4390848000 3293136000 1317254400 307359360 43908480 3920400 217800 7260 132 1 The maxiumum of lah(100, _) is: 44519005448993144810881324947684737529186447692709328597242209638906324913313742508392928375354932241404408343800007105650554669129521241784320000000000000000000000
Julia
<lang julia>using Combinatorics
function lah(n::Integer, k::Integer, signed=false)
if n == 0 || k == 0 || k > n
return zero(n)
elseif n == k
return one(n)
elseif k == 1
return factorial(n)
else
unsignedvalue = binomial(n, k) * binomial(n - 1, k - 1) * factorial(n - k)
if signed && isodd(n)
return -unsignedvalue
else
return unsignedvalue
end
end
end
function printlahtable(kmax)
println(" ", mapreduce(i -> lpad(i, 12), *, 0:kmax))
sstring(n, k) = begin i = lah(n, k); lpad(k > n && i == 0 ? "" : i, 12) end
for n in 0:kmax
println(rpad(n, 2) * mapreduce(k -> sstring(n, k), *, 0:kmax))
end
end
printlahtable(12)
println("\nThe maxiumum of lah(100, _) is: ", maximum(k -> lah(BigInt(100), BigInt(k)), 1:100))
</lang>
- Output:
0 1 2 3 4 5 6 7 8 9 10 11 12 0 0 1 0 1 2 0 2 1 3 0 6 6 1 4 0 24 36 12 1 5 0 120 240 120 20 1 6 0 720 1800 1200 300 30 1 7 0 5040 15120 12600 4200 630 42 1 8 0 40320 141120 141120 58800 11760 1176 56 1 9 0 362880 1451520 1693440 846720 211680 28224 2016 72 1 10 0 3628800 16329600 21772800 12700800 3810240 635040 60480 3240 90 1 11 0 39916800 199584000 299376000 199584000 69854400 13970880 1663200 118800 4950 110 1 12 0 479001600 2634508800 4390848000 3293136000 1317254400 307359360 43908480 3920400 217800 7260 132 1 The maxiumum of lah(100, _) is: 44519005448993144810881324947684737529186447692709328597242209638906324913313742508392928375354932241404408343800007105650554669129521241784320000000000000000000000
Kotlin
<lang scala>import java.math.BigInteger
fun factorial(n: BigInteger): BigInteger {
if (n == BigInteger.ZERO) return BigInteger.ONE
if (n == BigInteger.ONE) return BigInteger.ONE
var prod = BigInteger.ONE
var num = n
while (num > BigInteger.ONE) {
prod *= num
num--
}
return prod
}
fun lah(n: BigInteger, k: BigInteger): BigInteger {
if (k == BigInteger.ONE) return factorial(n) if (k == n) return BigInteger.ONE if (k > n) return BigInteger.ZERO if (k < BigInteger.ONE || n < BigInteger.ONE) return BigInteger.ZERO return (factorial(n) * factorial(n - BigInteger.ONE)) / (factorial(k) * factorial(k - BigInteger.ONE)) / factorial(n - k)
}
fun main() {
println("Unsigned Lah numbers: L(n, k):")
print("n/k ")
for (i in 0..12) {
print("%10d ".format(i))
}
println()
for (row in 0..12) {
print("%-3d".format(row))
for (i in 0..row) {
val l = lah(BigInteger.valueOf(row.toLong()), BigInteger.valueOf(i.toLong()))
print("%11d".format(l))
}
println()
}
println("\nMaximum value from the L(100, *) row:")
println((0..100).map { lah(BigInteger.valueOf(100.toLong()), BigInteger.valueOf(it.toLong())) }.max())
}</lang>
- Output:
Unsigned Lah numbers: L(n, k): n/k 0 1 2 3 4 5 6 7 8 9 10 11 12 0 1 1 0 1 2 0 2 1 3 0 6 6 1 4 0 24 36 12 1 5 0 120 240 120 20 1 6 0 720 1800 1200 300 30 1 7 0 5040 15120 12600 4200 630 42 1 8 0 40320 141120 141120 58800 11760 1176 56 1 9 0 362880 1451520 1693440 846720 211680 28224 2016 72 1 10 0 3628800 16329600 21772800 12700800 3810240 635040 60480 3240 90 1 11 0 39916800 199584000 299376000 199584000 69854400 13970880 1663200 118800 4950 110 1 12 0 479001600 2634508800 4390848000 3293136000 1317254400 307359360 43908480 3920400 217800 7260 132 1 Maximum value from the L(100, *) row: 44519005448993144810881324947684737529186447692709328597242209638906324913313742508392928375354932241404408343800007105650554669129521241784320000000000000000000000
Mathematica / Wolfram Language
<lang Mathematica>ClearAll[Lah] Lah[n_?Positive, 0] := 0 Lah[0, k_?Positive] := 0 Lah[n_, n_] := 1 Lah[n_, 1] := n! Lah[n_, k_] := (-1)^n (n! ((n - 1)!))/((k! ((k - 1)!)) ((n - k)!)) Table[Lah[i, j], {i, 0, 12}, {j, 0, 12}] // Grid Max[Lah[100, Range[0, 100]]]</lang>
- Output:
1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 2 1 0 0 0 0 0 0 0 0 0 0 0 6 -6 1 0 0 0 0 0 0 0 0 0 0 24 36 12 1 0 0 0 0 0 0 0 0 0 120 -240 -120 -20 1 0 0 0 0 0 0 0 0 720 1800 1200 300 30 1 0 0 0 0 0 0 0 5040 -15120 -12600 -4200 -630 -42 1 0 0 0 0 0 0 40320 141120 141120 58800 11760 1176 56 1 0 0 0 0 0 362880 -1451520 -1693440 -846720 -211680 -28224 -2016 -72 1 0 0 0 0 3628800 16329600 21772800 12700800 3810240 635040 60480 3240 90 1 0 0 0 39916800 -199584000 -299376000 -199584000 -69854400 -13970880 -1663200 -118800 -4950 -110 1 0 0 479001600 2634508800 4390848000 3293136000 1317254400 307359360 43908480 3920400 217800 7260 132 1 44519005448993144810881324947684737529186447692709328597242209638906324913313742508392928375354932241404408343800007105650554669129521241784320000000000000000000000
Nim
<lang Nim>import math, strutils import bignum
func lah[T: int | Int](n, k: T; signed = false): T =
if n == 0 or k == 0 or k > n: return when T is int: 0 else: newInt(0) if n == k: return when T is int: 1 else: newInt(1) if k == 1: return fac(n) result = binom(n, k) * binom(n - 1, k - 1) * fac(n - k) if signed and (n and 1) != 0: result = -result
proc printLahTable(kmax: int) =
stdout.write " "
for k in 0..kmax:
stdout.write ($k).align(12)
stdout.write('\n')
for n in 0..kmax:
stdout.write ($n).align(2)
for k in 0..n:
stdout.write ($lah(n, k)).align(12)
stdout.write('\n')
printLahTable(12)
var maxval = newInt(0) let n = newInt(100) for k in newInt(0)..newInt(100):
let val = lah(n, k) if val > maxval: maxval = val
echo "\nThe maximum value of lah(100, k) is ", maxval</lang>
- Output:
0 1 2 3 4 5 6 7 8 9 10 11 12 0 0 1 0 1 2 0 2 1 3 0 6 6 1 4 0 24 36 12 1 5 0 120 240 120 20 1 6 0 720 1800 1200 300 30 1 7 0 5040 15120 12600 4200 630 42 1 8 0 40320 141120 141120 58800 11760 1176 56 1 9 0 362880 1451520 1693440 846720 211680 28224 2016 72 1 10 0 3628800 16329600 21772800 12700800 3810240 635040 60480 3240 90 1 11 0 39916800 199584000 299376000 199584000 69854400 13970880 1663200 118800 4950 110 1 12 0 479001600 2634508800 4390848000 3293136000 1317254400 307359360 43908480 3920400 217800 7260 132 1 The maximum value of lah(100, k) is 44519005448993144810881324947684737529186447692709328597242209638906324913313742508392928375354932241404408343800007105650554669129521241784320000000000000000000000
Perl
<lang perl>use strict; use warnings; use feature 'say'; use ntheory qw(factorial); use List::Util qw(max);
sub Lah {
my($n, $k) = @_; return factorial($n) if $k == 1; return 1 if $k == $n; return 0 if $k > $n; return 0 if $k < 1 or $n < 1; (factorial($n) * factorial($n - 1)) / (factorial($k) * factorial($k - 1)) / factorial($n - $k)
}
my $upto = 12; my $mx = 1 + length max map { Lah(12,$_) } 0..$upto;
say 'Unsigned Lah numbers: L(n, k):'; print 'n\k' . sprintf "%${mx}s"x(1+$upto)."\n", 0..1+$upto;
for my $row (0..$upto) {
printf '%-3d', $row;
map { printf "%${mx}d", Lah($row, $_) } 0..$row;
print "\n";
}
say "\nMaximum value from the L(100, *) row:"; say max map { Lah(100,$_) } 0..100;</lang>
- Output:
Unsigned Lah numbers: L(n, k): n\k 0 1 2 3 4 5 6 7 8 9 10 11 0 1 1 0 1 2 0 2 1 3 0 6 6 1 4 0 24 36 12 1 5 0 120 240 120 20 1 6 0 720 1800 1200 300 30 1 7 0 5040 15120 12600 4200 630 42 1 8 0 40320 141120 141120 58800 11760 1176 56 1 9 0 362880 1451520 1693440 846720 211680 28224 2016 72 1 10 0 3628800 16329600 21772800 12700800 3810240 635040 60480 3240 90 1 11 0 39916800 199584000 299376000 199584000 69854400 13970880 1663200 118800 4950 110 1 12 0 479001600 2634508800 4390848000 3293136000 1317254400 307359360 43908480 3920400 217800 7260 132 Maximum value from the L(100, *) row: 44519005448993144810881324947684737529186447692709328597242209638906324913313742508392928375354932241404408343800007105650554669129521241784320000000000000000000000
Phix
with javascript_semantics include mpfr.e constant lim = 100, lim1 = lim+1, last = 12 sequence l = repeat(0,lim1) for n=1 to lim1 do l[n] = mpz_inits(lim1) mpz_set_si(l[n][n],1) if n!=2 then mpz_fac_ui(l[n][2],n-1) end if end for mpz {t, m100} = mpz_inits(2) for n=1 to lim do for k=1 to n do mpz_mul(t,l[n+1][2],l[n][2]) mpz_fdiv_q(t, t, l[k+1][2]) mpz_fdiv_q(t, t, l[k][2]) mpz_fdiv_q(l[n+1][k+1], t, l[n-k+1][2]) end for end for printf(1,"Unsigned Lah numbers: l(n, k):\n n k:") for i=0 to last do printf(1,"%6d ", i) end for printf(1,"\n--- %s\n",repeat('-',last*11+6)) for n=0 to last do printf(1,"%2d ", n) for k=1 to n+1 do printf(1,"%10s ",{mpz_get_str(l[n+1][k])}) end for printf(1,"\n") end for for k=1 to lim1 do mpz l100k = l[lim1][k] if mpz_cmp(l100k,m100) > 0 then mpz_set(m100,l100k) end if end for printf(1,"\nThe maximum l(100,k): %s\n",shorten(mpz_get_str(m100)))
- Output:
Unsigned Lah numbers: l(n, k): n k: 0 1 2 3 4 5 6 7 8 9 10 11 12 --- ------------------------------------------------------------------------------------------------------------------------------------------ 0 1 1 0 1 2 0 2 1 3 0 6 6 1 4 0 24 36 12 1 5 0 120 240 120 20 1 6 0 720 1800 1200 300 30 1 7 0 5040 15120 12600 4200 630 42 1 8 0 40320 141120 141120 58800 11760 1176 56 1 9 0 362880 1451520 1693440 846720 211680 28224 2016 72 1 10 0 3628800 16329600 21772800 12700800 3810240 635040 60480 3240 90 1 11 0 39916800 199584000 299376000 199584000 69854400 13970880 1663200 118800 4950 110 1 12 0 479001600 2634508800 4390848000 3293136000 1317254400 307359360 43908480 3920400 217800 7260 132 1 The maximum l(100,k): 4451900544899314481...0000000000000000000 (164 digits)
PicoLisp
<lang PicoLisp>(de fact (N)
(if (=0 N)
1
(* N (fact (dec N))) ) )
(de lah (N K)
(cond
((=1 K) (fact N))
((= N K) 1)
((> K N) 0)
((or (> 1 N) (> 1 K)) 0)
(T
(/
(* (fact N) (fact (dec N)))
(* (fact K) (fact (dec K)))
(fact (- N K)) ) ) ) )
(prin (align -12 "n/k")) (apply tab (range 0 12) (need 13 -11)) (for A (range 0 12)
(prin (align -2 A))
(for B (range 0 A)
(prin (align 11 (lah A B))) )
(prinl) )
(prinl "Maximum value from the L(100, *) row:") (maxi '((N) (lah 100 N)) (range 0 100)) (prinl @@)</lang>
- Output:
n/k 0 1 2 3 4 5 6 7 8 9 10 11 12 0 1 1 0 1 2 0 2 1 3 0 6 6 1 4 0 24 36 12 1 5 0 120 240 120 20 1 6 0 720 1800 1200 300 30 1 7 0 5040 15120 12600 4200 630 42 1 8 0 40320 141120 141120 58800 11760 1176 56 1 9 0 362880 1451520 1693440 846720 211680 28224 2016 72 1 10 0 3628800 16329600 21772800 12700800 3810240 635040 60480 3240 90 1 11 0 39916800 199584000 299376000 199584000 69854400 13970880 1663200 118800 4950 110 1 12 0 479001600 2634508800 4390848000 3293136000 1317254400 307359360 43908480 3920400 217800 7260 132 1 Maximum value from the L(100, *) row: 44519005448993144810881324947684737529186447692709328597242209638906324913313742508392928375354932241404408343800007105650554669129521241784320000000000000000000000
Prolog
<lang prolog>% Reference: https://en.wikipedia.org/wiki/Lah_number#Identities_and_relations
- - dynamic unsigned_lah_number_cache/3.
unsigned_lah_number(N, N, 1):-!. unsigned_lah_number(_, 0, 0):-!. unsigned_lah_number(N, K, 0):- K > N, !. unsigned_lah_number(N, K, L):- unsigned_lah_number_cache(N, K, L), !. unsigned_lah_number(N, K, L):- N1 is N - 1, K1 is K - 1, unsigned_lah_number(N1, K, L1), unsigned_lah_number(N1, K1, L2), !, L is (N1 + K) * L1 + L2, assertz(unsigned_lah_number_cache(N, K, L)).
print_unsigned_lah_numbers(N):- between(1, N, K), unsigned_lah_number(N, K, L), writef('%11r', [L]), fail. print_unsigned_lah_numbers(_):- nl.
print_unsigned_lah_numbers:- between(1, 12, N), print_unsigned_lah_numbers(N), fail. print_unsigned_lah_numbers.
max_unsigned_lah_number(N, Max):-
aggregate_all(max(L), (between(1, N, K), unsigned_lah_number(N, K, L)), Max).
main:- writeln('Unsigned Lah numbers up to L(12,12):'), print_unsigned_lah_numbers, writeln('Maximum value of L(n,k) where n = 100:'), max_unsigned_lah_number(100, M), writeln(M).</lang>
- Output:
Unsigned Lah numbers up to L(12,12):
1
2 1
6 6 1
24 36 12 1
120 240 120 20 1
720 1800 1200 300 30 1
5040 15120 12600 4200 630 42 1
40320 141120 141120 58800 11760 1176 56 1
362880 1451520 1693440 846720 211680 28224 2016 72 1
3628800 16329600 21772800 12700800 3810240 635040 60480 3240 90 1
39916800 199584000 299376000 199584000 69854400 13970880 1663200 118800 4950 110 1
479001600 2634508800 4390848000 3293136000 1317254400 307359360 43908480 3920400 217800 7260 132 1
Maximum value of L(n,k) where n = 100:
44519005448993144810881324947684737529186447692709328597242209638906324913313742508392928375354932241404408343800007105650554669129521241784320000000000000000000000
Python
<lang python>def factorial(n):
if n == 0:
return 1
res = 1
while n > 0:
res *= n
n -= 1
return res
def lah(n,k):
if k == 1:
return factorial(n)
if k == n:
return 1
if k > n:
return 0
if k < 1 or n < 1:
return 0
return (factorial(n) * factorial(n - 1)) / (factorial(k) * factorial(k - 1)) / factorial(n - k)
def main():
print "Unsigned Lah numbers: L(n, k):"
print "n/k ",
for i in xrange(13):
print "%11d" % i,
print
for row in xrange(13):
print "%-4d" % row,
for i in xrange(row + 1):
l = lah(row, i)
print "%11d" % l,
print
print "\nMaximum value from the L(100, *) row:"
maxVal = max([lah(100, a) for a in xrange(100)])
print maxVal
main()</lang>
- Output:
Unsigned Lah numbers: L(n, k): n/k 0 1 2 3 4 5 6 7 8 9 10 11 12 0 1 1 0 1 2 0 2 1 3 0 6 6 1 4 0 24 36 12 1 5 0 120 240 120 20 1 6 0 720 1800 1200 300 30 1 7 0 5040 15120 12600 4200 630 42 1 8 0 40320 141120 141120 58800 11760 1176 56 1 9 0 362880 1451520 1693440 846720 211680 28224 2016 72 1 10 0 3628800 16329600 21772800 12700800 3810240 635040 60480 3240 90 1 11 0 39916800 199584000 299376000 199584000 69854400 13970880 1663200 118800 4950 110 1 12 0 479001600 2634508800 4390848000 3293136000 1317254400 307359360 43908480 3920400 217800 7260 132 1 Maximum value from the L(100, *) row: 44519005448993144810881324947684737529186447692709328597242209638906324913313742508392928375354932241404408343800007105650554669129521241784320000000000000000000000
Quackery
<lang Quackery> [ table ] is ! ( n --> n )
1
101 times
[ dup ' ! put
i^ 1+ * ]
drop
[ 2dup = iff [ 2drop 1 ] done 2dup * 0 = iff * done dup 1 = iff [ drop ! ] done 2dup - ! unrot dup ! swap 1 - ! * swap dup ! swap 1 - ! * swap / swap / ] is lah ( n --> n )
[ dip number$ over size - space swap of swap join echo$ ] is justify ( n n --> )
[ table ] is colwidth ( n --> n )
13 times
[ 12 i^ lah number$
size 2 + ' colwidth put ]
say " k| "
13 times
[ i^ dup colwidth justify ] cr
say " n |"
char - 115 of echo$ cr
13 times
[ i^ dup
dup 2 justify say " | "
1+ times
[ dup i^ lah
i^ colwidth justify ]
drop cr ]
cr
0
101 times
[ 100 i^ lah
2dup < iff nip else drop ]
echo</lang>
- Output:
k| 0 1 2 3 4 5 6 7 8 9 10 11 12 n |------------------------------------------------------------------------------------------------------------------- 0 | 1 1 | 0 1 2 | 0 2 1 3 | 0 6 6 1 4 | 0 24 36 12 1 5 | 0 120 240 120 20 1 6 | 0 720 1800 1200 300 30 1 7 | 0 5040 15120 12600 4200 630 42 1 8 | 0 40320 141120 141120 58800 11760 1176 56 1 9 | 0 362880 1451520 1693440 846720 211680 28224 2016 72 1 10 | 0 3628800 16329600 21772800 12700800 3810240 635040 60480 3240 90 1 11 | 0 39916800 199584000 299376000 199584000 69854400 13970880 1663200 118800 4950 110 1 12 | 0 479001600 2634508800 4390848000 3293136000 1317254400 307359360 43908480 3920400 217800 7260 132 1 44519005448993144810881324947684737529186447692709328597242209638906324913313742508392928375354932241404408343800007105650554669129521241784320000000000000000000000
R
<lang R>Lah_numbers <- function(n, k, type = "unsigned") {
if (n == k)
return(1)
if (n == 0 | k == 0)
return(0)
if (k == 1)
return(factorial(n))
if (k > n)
return(NA)
if (type == "unsigned")
return((factorial(n) * factorial(n - 1)) / (factorial(k) * factorial(k - 1)) / factorial(n - k))
if (type == "signed")
return(-1 ** n * (factorial(n) * factorial(n - 1)) / (factorial(k) * factorial(k - 1)) / factorial(n - k))
}
- demo
Table <- matrix(0 , 13, 13, dimnames = list(0:12, 0:12))
for (n in 0:12) {
for (k in 0:12) {
Table[n + 1, k + 1] <- Lah_numbers(n, k, type = "unsigned")
}
}
Table
</lang>
- Output:
0 1 2 3 4 5 6 7 8 9 10 11 12 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 NA NA NA NA NA NA NA NA NA NA NA 2 0 2 1 NA NA NA NA NA NA NA NA NA NA 3 0 6 6 1 NA NA NA NA NA NA NA NA NA 4 0 24 36 12 1 NA NA NA NA NA NA NA NA 5 0 120 240 120 20 1 NA NA NA NA NA NA NA 6 0 720 1800 1200 300 30 1 NA NA NA NA NA NA 7 0 5040 15120 12600 4200 630 42 1 NA NA NA NA NA 8 0 40320 141120 141120 58800 11760 1176 56 1 NA NA NA NA 9 0 362880 1451520 1693440 846720 211680 28224 2016 72 1 NA NA NA 10 0 3628800 16329600 21772800 12700800 3810240 635040 60480 3240 90 1 NA NA 11 0 39916800 199584000 299376000 199584000 69854400 13970880 1663200 118800 4950 110 1 NA 12 0 479001600 2634508800 4390848000 3293136000 1317254400 307359360 43908480 3920400 217800 7260 132 1
Raku
(formerly Perl 6)
<lang perl6>constant @factorial = 1, |[\*] 1..*;
sub Lah (Int \n, Int \k) {
return @factorial[n] if k == 1; return 1 if k == n; return 0 if k > n; return 0 if k < 1 or n < 1; (@factorial[n] * @factorial[n - 1]) / (@factorial[k] * @factorial[k - 1]) / @factorial[n - k]
}
my $upto = 12;
my $mx = (1..$upto).map( { Lah($upto, $_) } ).max.chars;
put 'Unsigned Lah numbers: L(n, k):'; put 'n\k', (0..$upto)».fmt: "%{$mx}d";
for 0..$upto -> $row {
$row.fmt('%-3d').print;
put (0..$row).map( { Lah($row, $_) } )».fmt: "%{$mx}d";
}
say "\nMaximum value from the L(100, *) row:"; say (^100).map( { Lah 100, $_ } ).max;</lang>
- Output:
Unsigned Lah numbers: L(n, k): n\k 0 1 2 3 4 5 6 7 8 9 10 11 12 0 1 1 0 1 2 0 2 1 3 0 6 6 1 4 0 24 36 12 1 5 0 120 240 120 20 1 6 0 720 1800 1200 300 30 1 7 0 5040 15120 12600 4200 630 42 1 8 0 40320 141120 141120 58800 11760 1176 56 1 9 0 362880 1451520 1693440 846720 211680 28224 2016 72 1 10 0 3628800 16329600 21772800 12700800 3810240 635040 60480 3240 90 1 11 0 39916800 199584000 299376000 199584000 69854400 13970880 1663200 118800 4950 110 1 12 0 479001600 2634508800 4390848000 3293136000 1317254400 307359360 43908480 3920400 217800 7260 132 1 Maximum value from the L(100, *) row: 44519005448993144810881324947684737529186447692709328597242209638906324913313742508392928375354932241404408343800007105650554669129521241784320000000000000000000000
REXX
Some extra code was added to minimize the column widths in the displaying of the numbers.
Also, code was added to use memoization of the factorial calculations. <lang rexx>/*REXX pgm computes & display (unsigned) Stirling numbers of the 3rd kind (Lah numbers).*/ parse arg lim . /*obtain optional argument from the CL.*/ if lim== | lim=="," then lim= 12 /*Not specified? Then use the default.*/ olim= lim /*save the original value of LIM. */ lim= abs(lim) /*only use the absolute value of LIM. */ numeric digits max(9, 4*lim) /*(over) specify maximum number in grid*/ max#.= 0 !.=. @.= /* [↓] calculate values for the grid. */
do n=0 to lim; nm= n - 1
do k=0 to lim; km= k - 1
if k==1 then do; @.n.k= !(n); call maxer; iterate; end
if k==n then do; @.n.k= 1 ; iterate; end
if k>n | k==0 | n==0 then do; @.n.k= 0 ; iterate; end
@.n.k = (!(n) * !(nm)) % (!(k) * !(km)) % !(n-k) /*calculate a # in the grid.*/
call maxer /*find max # " " " */
end /*k*/
end /*n*/
do k=0 for lim+1 /*find max column width for each column*/
max#.a= max#.a + length(max#.k)
end /*k*/
/* [↓] only show the maximum value ? */
w= length(max#.b) /*calculate max width of all numbers. */ if olim<0 then do; say 'The maximum value (which has ' w " decimal digits):"
say max#.b /*display maximum number in the grid. */
exit /*stick a fork in it, we're all done. */
end /* [↑] the 100th row is when LIM is 99*/
wi= max(3, length(lim+1) ) /*the maximum width of the grid's index*/ say 'row' center('columns', max(9, max#.a + lim), '═') /*display header of the grid.*/
do r=0 for lim+1; $= /* [↓] display the grid to the term. */
do c=0 for lim+1 until c>=r /*build a row of grid, 1 col at a time.*/
$= $ right(@.r.c, length(max#.c) ) /*append a column to a row of the grid.*/
end /*c*/
say right(r,wi) strip(substr($,2), 'T') /*display a single row of the grid. */
end /*r*/
exit /*stick a fork in it, we're all done. */ /*──────────────────────────────────────────────────────────────────────────────────────*/ !: parse arg z; if !.z\==. then return !.z; !=1; do f=2 to z; !=!*f; end; !.z=!; return ! maxer: max#.k= max(max#.k, @.n.k); max#.b= max(max#.b, @.n.k); return</lang>
- output when using the default input:
row ══════════════════════════════════════════════columns═══════════════════════════════════════════════ 0 1 1 0 1 2 0 2 1 3 0 6 6 1 4 0 24 36 12 1 5 0 120 240 120 20 1 6 0 720 1800 1200 300 30 1 7 0 5040 15120 12600 4200 630 42 1 8 0 40320 141120 141120 58800 11760 1176 56 1 9 0 362880 1451520 1693440 846720 211680 28224 2016 72 1 10 0 3628800 16329600 21772800 12700800 3810240 635040 60480 3240 90 1 11 0 39916800 199584000 299376000 199584000 69854400 13970880 1663200 118800 4950 110 1 12 0 479001600 2634508800 4390848000 3293136000 1317254400 307359360 43908480 3920400 217800 7260 132 1
- output when using the input of: -100
The maximum value (which has 164 decimal digits): 44519005448993144810881324947684737529186447692709328597242209638906324913313742508392928375354932241404408343800007105650554669129521241784320000000000000000000000
Ruby
Works with Ruby 3.0 (end-less method; end-less and begin-less range). <lang ruby>def fact(n) = n.zero? ? 1 : 1.upto(n).inject(&:*)
def lah(n, k)
case k when 1 then fact(n) when n then 1 when (..1),(n..) then 0 else n<1 ? 0 : (fact(n)*fact(n-1)) / (fact(k)*fact(k-1)) / fact(n-k) end
end
r = (0..12) puts "Unsigned Lah numbers: L(n, k):" puts "n/k #{r.map{|n| "%11d" % n}.join}"
r.each do |row|
print "%-4s" % row
puts "#{(0..row).map{|col| "%11d" % lah(row,col)}.join}"
end
puts "\nMaximum value from the L(100, *) row:"; puts (1..100).map{|a| lah(100,a)}.max </lang>
- Output:
Unsigned Lah numbers: L(n, k): n/k 0 1 2 3 4 5 6 7 8 9 10 11 12 0 1 1 0 1 2 0 2 1 3 0 6 6 1 4 0 24 36 12 1 5 0 120 240 120 20 1 6 0 720 1800 1200 300 30 1 7 0 5040 15120 12600 4200 630 42 1 8 0 40320 141120 141120 58800 11760 1176 56 1 9 0 362880 1451520 1693440 846720 211680 28224 2016 72 1 10 0 3628800 16329600 21772800 12700800 3810240 635040 60480 3240 90 1 11 0 39916800 199584000 299376000 199584000 69854400 13970880 1663200 118800 4950 110 1 12 0 479001600 2634508800 4390848000 3293136000 1317254400 307359360 43908480 3920400 217800 7260 132 1 Maximum value from the L(100, *) row: 44519005448993144810881324947684737529186447692709328597242209638906324913313742508392928375354932241404408343800007105650554669129521241784320000000000000000000000
Scheme
<lang scheme>; Compute the Unsigned Lah number L(n, k). (define lah
(lambda (n k) (/ (/ (* (fact n) (fact (1- n))) (* (fact k) (fact (1- k)))) (fact (- n k)))))
- Procedure to compute factorial.
(define fact
(lambda (n)
(if (<= n 0)
1
(* n (fact (1- n))))))
- Generate a table of the Unsigned Lah numbers L(n, k) up to L(12, 12).
(printf "The Unsigned Lah numbers L(n, k) up to L(12, 12):~%") (printf "n\\k~10d" 1) (do ((k 2 (1+ k)))
((> k 12)) (printf " ~10d" k))
(newline) (do ((n 1 (1+ n)))
((> n 12))
(printf "~2d" n)
(do ((k 1 (1+ k)))
((> k n))
(printf " ~10d" (lah n k)))
(newline))
- Find the maximum value of L(n, k) where n = 100.
(printf "~%The maximum value of L(n, k) where n = 100:~%") (let ((max 0))
(do ((k 1 (1+ k)))
((> k 100))
(let ((val (lah 100 k)))
(when (> val max) (set! max val))))
(printf "~d~%" max))</lang>
- Output:
The Unsigned Lah numbers L(n, k) up to L(12, 12): n\k 1 2 3 4 5 6 7 8 9 10 11 12 1 1 2 2 1 3 6 6 1 4 24 36 12 1 5 120 240 120 20 1 6 720 1800 1200 300 30 1 7 5040 15120 12600 4200 630 42 1 8 40320 141120 141120 58800 11760 1176 56 1 9 362880 1451520 1693440 846720 211680 28224 2016 72 1 10 3628800 16329600 21772800 12700800 3810240 635040 60480 3240 90 1 11 39916800 199584000 299376000 199584000 69854400 13970880 1663200 118800 4950 110 1 12 479001600 2634508800 4390848000 3293136000 1317254400 307359360 43908480 3920400 217800 7260 132 1 The maximum value of L(n, k) where n = 100: 44519005448993144810881324947684737529186447692709328597242209638906324913313742508392928375354932241404408343800007105650554669129521241784320000000000000000000000
Sidef
<lang ruby>func lah(n, k) {
stirling3(n, k) #binomial(n-1, k-1) * n!/k! # alternative formula
}
const r = (0..12)
var triangle = r.map {|n| 0..n -> map {|k| lah(n, k) } } var widths = r.map {|n| r.map {|k| (triangle[k][n] \\ 0).len }.max }
say ('n\k ', r.map {|n| "%*s" % (widths[n], n) }.join(' '))
r.each {|n|
var str = ('%-3s ' % n)
str += triangle[n].map_kv {|k,v| "%*s" % (widths[k], v) }.join(' ')
say str
}
with (100) {|n|
say "\nMaximum value from the L(#{n}, *) row:"
say { lah(n, _) }.map(^n).max
}</lang>
- Output:
n\k 0 1 2 3 4 5 6 7 8 9 10 11 12 0 1 1 0 1 2 0 2 1 3 0 6 6 1 4 0 24 36 12 1 5 0 120 240 120 20 1 6 0 720 1800 1200 300 30 1 7 0 5040 15120 12600 4200 630 42 1 8 0 40320 141120 141120 58800 11760 1176 56 1 9 0 362880 1451520 1693440 846720 211680 28224 2016 72 1 10 0 3628800 16329600 21772800 12700800 3810240 635040 60480 3240 90 1 11 0 39916800 199584000 299376000 199584000 69854400 13970880 1663200 118800 4950 110 1 12 0 479001600 2634508800 4390848000 3293136000 1317254400 307359360 43908480 3920400 217800 7260 132 1 Maximum value from the L(100, *) row: 44519005448993144810881324947684737529186447692709328597242209638906324913313742508392928375354932241404408343800007105650554669129521241784320000000000000000000000
Swift
<lang swift>import BigInt import Foundation
@inlinable public func factorial<T: BinaryInteger>(_ n: T) -> T {
guard n != 0 else {
return 1
}
return stride(from: n, to: 0, by: -1).reduce(1, *)
}
@inlinable public func lah<T: BinaryInteger>(n: T, k: T) -> T {
if k == 1 {
return factorial(n)
} else if k == n {
return 1
} else if k > n {
return 0
} else if k < 1 || n < 1 {
return 0
} else {
let a = (factorial(n) * factorial(n - 1))
let b = (factorial(k) * factorial(k - 1))
let c = factorial(n - k)
return a / b / c }
}
print("Unsigned Lah numbers: L(n, k):") print("n\\k", terminator: "")
for i in 0...12 {
print(String(format: "%10d", i), terminator: " ")
}
print()
for row in 0...12 {
print(String(format: "%-2d", row), terminator: "")
for i in 0...row {
lah(n: BigInt(row), k: BigInt(i)).description.withCString {str in
print(String(format: "%11s", str), terminator: "")
}
}
print()
}
let maxLah = (0...100).map({ lah(n: BigInt(100), k: BigInt($0)) }).max()!
print("Maximum value from the L(100, *) row: \(maxLah)")</lang>
- Output:
Unsigned Lah numbers: L(n, k): n\k 0 1 2 3 4 5 6 7 8 9 10 11 12 0 1 1 0 1 2 0 2 1 3 0 6 6 1 4 0 24 36 12 1 5 0 120 240 120 20 1 6 0 720 1800 1200 300 30 1 7 0 5040 15120 12600 4200 630 42 1 8 0 40320 141120 141120 58800 11760 1176 56 1 9 0 362880 1451520 1693440 846720 211680 28224 2016 72 1 10 0 3628800 16329600 21772800 12700800 3810240 635040 60480 3240 90 1 11 0 39916800 199584000 299376000 199584000 69854400 13970880 1663200 118800 4950 110 1 12 0 479001600 2634508800 4390848000 3293136000 1317254400 307359360 43908480 3920400 217800 7260 132 1 Maximum value from the L(100, *) row: 44519005448993144810881324947684737529186447692709328597242209638906324913313742508392928375354932241404408343800007105650554669129521241784320000000000000000000000
Tcl
<lang Tcl>proc prod {from to} {
set r 1
if {$from <= $to} {
set r $from
while {[incr from] <= $to} {
set r [expr {$r * $from}]
}
}
return $r
}
proc US3 {n k} {
if {$n < 0 || $k < 0} {
error "US3(): negative arg ($n,$k)"
}
## L(0,0) = 1
## L(n,0) = 0 if 0 < n
## L(0,k) = 0 if 0 < k
## L(n,k) = 0 if n < k
## L(n,n) = 1
if {$n == $k} {
return 1
}
if {$n == 0 || $k == 0} {
return 0
}
if {$n < $k} {
return 0
}
set nk [list $n $k]
if {[info exists ::US3cache($nk)]} {
return $::US3cache($nk)
}
if {$k == 1} {
## L(n,1) = n!
set r [prod 2 $n]
} else {
## k > 1
## L(n,k) = L(n,k-1) * (n - (k-1)) / ((k-1)*k)
set k1 [expr {$k - 1}]
set r [expr {([US3 $n $k1] * ($n - $k1)) / ($k * $k1)}]
}
set ::US3cache($nk) $r
}
proc main {} {
puts "Unsigned Lah numbers L(n,k):"
set max 12 ;# last n,k to print
set L 10 ;# space to use for 1 number
set minn 1 ;# first row to print
set mink 1 ;# first column to print
puts -nonewline "n\\k"
for {set n $minn} {$n <= $max} {incr n} {
puts -nonewline " [format %${L}d $n]"
}
puts ""
for {set n $minn} {$n <= $max} {incr n} {
puts -nonewline [format %3d $n]
for {set k $mink} {$k <= $n} {incr k} {
puts -nonewline " [format %${L}s [US3 $n $k]]"
}
puts ""
}
set n 100
puts "The maximum value of L($n, k) = "
set maxv 0
set maxk -1
for {set k 0} {$k <= $n} {incr k} {
set v [US3 $n $k]
if {$v > $maxv} {
set maxv $v
set maxk $k
}
}
puts $maxv
puts "([string length $maxv] digits, k=$maxk)"
} main </lang>
- Output:
Unsigned Lah numbers L(n,k): n\k 1 2 3 4 5 6 7 8 9 10 11 12 1 1 2 2 1 3 6 6 1 4 24 36 12 1 5 120 240 120 20 1 6 720 1800 1200 300 30 1 7 5040 15120 12600 4200 630 42 1 8 40320 141120 141120 58800 11760 1176 56 1 9 362880 1451520 1693440 846720 211680 28224 2016 72 1 10 3628800 16329600 21772800 12700800 3810240 635040 60480 3240 90 1 11 39916800 199584000 299376000 199584000 69854400 13970880 1663200 118800 4950 110 1 12 479001600 2634508800 4390848000 3293136000 1317254400 307359360 43908480 3920400 217800 7260 132 1 The maximum value of L(100, k) = 44519005448993144810881324947684737529186447692709328597242209638906324913313742508392928375354932241404408343800007105650554669129521241784320000000000000000000000 (164 digits, k=10)
VBScript
<lang vb>' Lah numbers - VBScript - 04/02/2021
Function F(i,n)
Dim c: c=CCur(i): If n>Len(c) Then F=Space(n-Len(c))&c Else F=c
End Function 'F
Function Fact(ByVal n)
Dim res
If n=0 Then
Fact = 1
Else
res = 1
While n>0
res = res*n
n = n-1
Wend
Fact = res
End If
End Function 'Fact
Function Lah(n, k)
If k=1 Then
Lah = Fact(n)
ElseIf k=n Then
Lah = 1
ElseIf k>n Then
Lah=0
ElseIf k < 1 Or n < 1 Then
Lah = 0
Else
Lah = (Fact(n) * Fact(n-1)) / (Fact(k) * Fact(k-1)) / Fact(n-k)
End If
End Function 'Lah
Sub Main()
ns=12: p=10
WScript.Echo "Unsigned Lah numbers: Lah(n,k):"
buf = "n/k "
For k=1 To ns
buf = buf & F(k,p) & " "
Next 'k
WScript.Echo buf
For n=1 To ns
buf = F(n,3) & " "
For k=1 To n
l = Lah(n,k)
buf = buf & F(l,p) & " "
Next 'k
WScript.Echo buf
Next 'n
End Sub 'Main
Main() </lang>
- Output:
Unsigned Lah numbers: Lah(n,k): n/k 1 2 3 4 5 6 7 8 9 10 11 12 1 1 2 2 1 3 6 6 1 4 24 36 12 1 5 120 240 120 20 1 6 720 1800 1200 300 30 1 7 5040 15120 12600 4200 630 42 1 8 40320 141120 141120 58800 11760 1176 56 1 9 362880 1451520 1693440 846720 211680 28224 2016 72 1 10 3628800 16329600 21772800 12700800 3810240 635040 60480 3240 90 1 11 39916800 199584000 299376000 199584000 69854400 13970880 1663200 118800 4950 110 1 12 479001600 2634508800 4390848000 3293136000 1317254400 307359360 43908480 3920400 217800 7260 132 1
Visual Basic .NET
<lang vbnet>Imports System.Numerics
Module Module1
Function Factorial(n As BigInteger) As BigInteger
If n = 0 Then
Return 1
End If
Dim res As BigInteger = 1
While n > 0
res *= n
n -= 1
End While
Return res
End Function
Function Lah(n As BigInteger, k As BigInteger) As BigInteger
If k = 1 Then
Return Factorial(n)
End If
If k = n Then
Return 1
End If
If k > n Then
Return 0
End If
If k < 1 OrElse n < 1 Then
Return 0
End If
Return (Factorial(n) * Factorial(n - 1)) / (Factorial(k) * Factorial(k - 1)) / Factorial(n - k)
End Function
Sub Main()
Console.WriteLine("Unsigned Lah numbers: L(n, k):")
Console.Write("n/k ")
For Each i In Enumerable.Range(0, 13)
Console.Write("{0,10} ", i)
Next
Console.WriteLine()
For Each row In Enumerable.Range(0, 13)
Console.Write("{0,-3}", row)
For Each i In Enumerable.Range(0, row + 1)
Dim l = Lah(row, i)
Console.Write("{0,11}", l)
Next
Console.WriteLine()
Next
Console.WriteLine()
Console.WriteLine("Maximum value from the L(100, *) row:")
Dim maxVal = Enumerable.Range(0, 100).Select(Function(a) Lah(100, a)).Max
Console.WriteLine(maxVal)
End Sub
End Module</lang>
- Output:
Unsigned Lah numbers: L(n, k): n/k 0 1 2 3 4 5 6 7 8 9 10 11 12 0 1 1 0 1 2 0 2 1 3 0 6 6 1 4 0 24 36 12 1 5 0 120 240 120 20 1 6 0 720 1800 1200 300 30 1 7 0 5040 15120 12600 4200 630 42 1 8 0 40320 141120 141120 58800 11760 1176 56 1 9 0 362880 1451520 1693440 846720 211680 28224 2016 72 1 10 0 3628800 16329600 21772800 12700800 3810240 635040 60480 3240 90 1 11 0 39916800 199584000 299376000 199584000 69854400 13970880 1663200 118800 4950 110 1 12 0 479001600 2634508800 4390848000 3293136000 1317254400 307359360 43908480 3920400 217800 7260 132 1 Maximum value from the L(100, *) row: 44519005448993144810881324947684737529186447692709328597242209638906324913313742508392928375354932241404408343800007105650554669129521241784320000000000000000000000
Wren
<lang ecmascript>import "/fmt" for Fmt
var fact = Fn.new { |n|
if (n < 2) return 1 var fact = 1 for (i in 2..n) fact = fact * i return fact
}
var lah = Fn.new { |n, k|
if (k == 1) return fact.call(n) if (k == n) return 1 if (k > n) return 0 if (k < 1 || n < 1) return 0 return (fact.call(n) * fact.call(n-1)) / (fact.call(k) * fact.call(k-1)) / fact.call(n-k)
}
System.print("Unsigned Lah numbers: l(n, k):") System.write("n/k") for (i in 0..12) System.write("%(Fmt.d(10, i)) ") System.print("\n" + "-" * 145) for (n in 0..12) {
System.write("%(Fmt.d(2, n)) ")
for (k in 0..n) System.write("%(Fmt.d(10, lah.call(n, k))) ")
System.print()
}</lang>
- Output:
Unsigned Lah numbers: l(n, k): n/k 0 1 2 3 4 5 6 7 8 9 10 11 12 ------------------------------------------------------------------------------------------------------------------------------------------------- 0 1 1 0 1 2 0 2 1 3 0 6 6 1 4 0 24 36 12 1 5 0 120 240 120 20 1 6 0 720 1800 1200 300 30 1 7 0 5040 15120 12600 4200 630 42 1 8 0 40320 141120 141120 58800 11760 1176 56 1 9 0 362880 1451520 1693440 846720 211680 28224 2016 72 1 10 0 3628800 16329600 21772800 12700800 3810240 635040 60480 3240 90 1 11 0 39916800 199584000 299376000 199584000 69854400 13970880 1663200 118800 4950 110 1 12 0 479001600 2634508800 4390848000 3293136000 1317254400 307359360 43908480 3920400 217800 7260 132 1
Vala
<lang Vala>uint64 factorial(uint8 n) {
uint64 res = 1; if (n == 0) return res; while (n > 0) res *= n--; return res;
}
uint64 lah(uint8 n, uint8 k) {
if (k == 1) return factorial(n); if (k == n) return 1; if (k > n) return 0; if (k < 1 || n < 1) return 0; return (factorial(n) * factorial(n - 1)) / (factorial(k) * factorial(k - 1)) / factorial(n - k);
}
void main() {
uint8 row, i;
print("Unsigned Lah numbers: L(n, k):\n");
print("n/k ");
for (i = 0; i < 13; i++) {
print("%10d ", i);
}
print("\n");
for (row = 0; row < 13; row++) {
print("%-3d", row);
for (i = 0; i < row + 1; i++) {
uint64 l = lah(row, i);
print("%11lld", l);
}
print("\n");
}
}</lang>
- Output:
Unsigned Lah numbers: L(n, k): n/k 0 1 2 3 4 5 6 7 8 9 10 11 12 0 1 1 0 1 2 0 2 1 3 0 6 6 1 4 0 24 36 12 1 5 0 120 240 120 20 1 6 0 720 1800 1200 300 30 1 7 0 5040 15120 12600 4200 630 42 1 8 0 40320 141120 141120 58800 11760 1176 56 1 9 0 362880 1451520 1693440 846720 211680 28224 2016 72 1 10 0 3628800 16329600 21772800 12700800 3810240 635040 60480 3240 90 1 11 0 39916800 199584000 299376000 199584000 69854400 13970880 1663200 118800 4950 110 1 12 0 479001600 2634508800 4390848000 3293136000 1317254400 307359360 43908480 3920400 217800 7260 132 1
zkl
<lang zkl>fcn lah(n,k,fact=fcn(n){ [1..n].reduce('*,1) }){
if(n==k) return(1); if(k==1) return(fact(n)); if(n<1 or k<1) return(0); (fact(n)*fact(n - 1)) /(fact(k)*fact(k - 1)) /fact(n - k)
}</lang> <lang zkl>// calculate entire table (quick), find max, find num digits in max N,mx := 12, [1..N].apply(fcn(n){ [1..n].apply(lah.fp(n)) }).flatten() : (0).max(_); fmt:="%%%dd".fmt("%d".fmt(mx.numDigits + 1)).fmt; // "%9d".fmt println("Unsigned Lah numbers: L(n,k):"); println("n\\k",[0..N].pump(String,fmt)); foreach row in ([0..N]){
println("%3d".fmt(row), [0..row].pump(String, lah.fp(row), fmt));
}</lang>
- Output:
Unsigned Lah numbers: L(n,k): n\k 0 1 2 3 4 5 6 7 8 9 10 11 12 0 0 1 0 1 2 0 2 1 3 0 6 6 1 4 0 24 36 12 1 5 0 120 240 120 20 1 6 0 720 1800 1200 300 30 1 7 0 5040 15120 12600 4200 630 42 1 8 0 40320 141120 141120 58800 11760 1176 56 1 9 0 362880 1451520 1693440 846720 211680 28224 2016 72 1 10 0 3628800 16329600 21772800 12700800 3810240 635040 60480 3240 90 1 11 0 39916800 199584000 299376000 199584000 69854400 13970880 1663200 118800 4950 110 1 12 0 479001600 2634508800 4390848000 3293136000 1317254400 307359360 43908480 3920400 217800 7260 132 1
GNU Multiple Precision Arithmetic Library
<lang zkl>var [const] BI=Import("zklBigNum"); // libGMP N=100; L100:=[1..N].apply(lah.fpM("101",BI(N),fcn(n){ BI(n).factorial() }))
.reduce(fcn(m,n){ m.max(n) });
println("Maximum value from the L(%d, *) row (%d digits):".fmt(N,L100.numDigits)); println(L100);</lang>
- Output:
Maximum value from the L(100, *) row (164 digits): 44519005448993144810881324947684737529186447692709328597242209638906324913313742508392928375354932241404408343800007105650554669129521241784320000000000000000000000