Lah numbers: Difference between revisions
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=={{header|zkl}}== |
=={{header|zkl}}== |
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<lang zkl>fcn lah(n,k,fact=fcn(n){ [1..n].reduce('*,1) }){ |
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if(n==0 or k==0) return(0); |
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if(n==k) return(1); |
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if(k==1) return(fact(n)); |
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(fact(n)*fact(n - 1)) /(fact(k)*fact(k - 1)) /fact(n - k) |
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}</lang> |
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<lang zkl>// calculate entire table (quick), find max, find num digits in max |
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N,mx := 12, [1..N].apply(fcn(n){ [1..n].apply(lah.fp(n)) }).flatten() : (0).max(_); |
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fmt:="%%%dd".fmt("%d".fmt(mx.numDigits + 1)).fmt; // "%9d".fmt |
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println("Unsigned Lah numbers: L(n,k):"); |
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println("n\\k",[0..N].pump(String,fmt)); |
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foreach row in ([0..N]){ |
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println("%3d".fmt(row), [0..row].pump(String, lah.fp(row), fmt)); |
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}</lang> |
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{{out}} |
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<pre style="font-size:83%"> |
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Unsigned Lah numbers: L(n,k): |
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n\k 0 1 2 3 4 5 6 7 8 9 10 11 12 |
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0 0 |
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1 0 1 |
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2 0 2 1 |
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3 0 6 6 1 |
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4 0 24 36 12 1 |
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5 0 120 240 120 20 1 |
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6 0 720 1800 1200 300 30 1 |
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7 0 5040 15120 12600 4200 630 42 1 |
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8 0 40320 141120 141120 58800 11760 1176 56 1 |
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9 0 362880 1451520 1693440 846720 211680 28224 2016 72 1 |
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10 0 3628800 16329600 21772800 12700800 3810240 635040 60480 3240 90 1 |
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11 0 39916800 199584000 299376000 199584000 69854400 13970880 1663200 118800 4950 110 1 |
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12 0 479001600 2634508800 4390848000 3293136000 1317254400 307359360 43908480 3920400 217800 7260 132 1 |
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</pre> |
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{{libheader|GMP}} GNU Multiple Precision Arithmetic Library |
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<lang zkl>var [const] BI=Import("zklBigNum"); // libGMP |
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N=100; |
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println("Maximum value from the L(%d, *) row:".fmt(N)); |
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[1..N].apply(lah.fpM("101",BI(N),fcn(n){ BI(n).factorial() })) |
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.reduce(fcn(m,n){ m.max(n) }).println();</lang> |
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{{out}} |
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<pre style="font-size:83%"> |
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Maximum value from the L(100, *) row: |
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44519005448993144810881324947684737529186447692709328597242209638906324913313742508392928375354932241404408343800007105650554669129521241784320000000000000000000000 |
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</pre> |
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Revision as of 18:50, 19 August 2019
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.
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
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
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.
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
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 displaying of the column widths.
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
zkl
<lang zkl>fcn lah(n,k,fact=fcn(n){ [1..n].reduce('*,1) }){
if(n==0 or k==0) return(0); if(n==k) return(1); if(k==1) return(fact(n)); (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; println("Maximum value from the L(%d, *) row:".fmt(N)); [1..N].apply(lah.fpM("101",BI(N),fcn(n){ BI(n).factorial() }))
.reduce(fcn(m,n){ m.max(n) }).println();</lang>
- Output:
Maximum value from the L(100, *) row: 44519005448993144810881324947684737529186447692709328597242209638906324913313742508392928375354932241404408343800007105650554669129521241784320000000000000000000000