⚠️ Warning: This is a draft ⚠️
This means it might contain formatting issues, incorrect code, conceptual problems, or other severe issues.
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{{draft task}} 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:
:* '''[[wp:Lah_number|Wikipedia - Lah number]]''' :* '''[[oeis:A105278|OEIS:A105278 - Unsigned Lah numbers]]''' :* '''[[oeis:A008297|OEIS:A008297 - Signed Lah numbers]]'''
;Related Tasks:
:* '''[[Stirling_numbers_of_the_first_kind|Stirling numbers of the first kind]]''' :* '''[[Stirling_numbers_of_the_second_kind|Stirling numbers of the second kind]]''' :* '''[[Bell_numbers|Bell numbers]]'''
D
{{trans|Kotlin}}
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);
}
{{out}}
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
{{works with|Factor|0.99 development version 2019-07-10}}
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 .
{{out}}
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
=={{header|Fōrmulæ}}==
In [https://wiki.formulae.org/Lah_numbers this] page you can see the solution of this task.
Fōrmulæ programs are not textual, visualization/edition of programs is done showing/manipulating structures but not text ([http://wiki.formulae.org/Editing_F%C5%8Drmul%C3%A6_expressions more info]). 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 transportation effects more than visualization and edition.
The option to show Fōrmulæ programs and their results is showing images. Unfortunately images cannot be uploaded in Rosetta Code.
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()))
}
{{out}}
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
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))
{{out}}
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
{{trans|Perl}}
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())
}
{{out}}
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
Perl
{{libheader|ntheory}} {{trans|Perl 6}}
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;
{{out}}
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
Perl 6
{{works with|Rakudo|2019.07.1}}
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;
{{out}}
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
Phix
{{libheader|mpfr}} {{trans|Go}}
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
mpfr_printf(1,"%10Zd ", 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)))
{{out}}
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)
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.
/*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
{{out|output|text= 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
{{out|output|text= when using the input of: -100 }}
The maximum value (which has 164 decimal digits):
44519005448993144810881324947684737529186447692709328597242209638906324913313742508392928375354932241404408343800007105650554669129521241784320000000000000000000000
Sidef
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
}
{{out}}
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
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)
}
// 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));
}
{{out}}
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
```
{{libheader|GMP}} GNU Multiple Precision Arithmetic Library
```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);
```
{{out}}
Maximum value from the L(100, *) row (164 digits):
44519005448993144810881324947684737529186447692709328597242209638906324913313742508392928375354932241404408343800007105650554669129521241784320000000000000000000000
```