Stirling numbers of the second kind

From Rosetta Code
Stirling numbers of the second kind is a draft programming task. It is not yet considered ready to be promoted as a complete task, for reasons that should be found in its talk page.

Stirling numbers of the second kind, or Stirling partition numbers, are the number of ways to partition a set of n objects into k non-empty subsets. They are closely related to Bell numbers, and may be derived from them.


Stirling numbers of the second kind obey the recurrence relation:

   S2(n, 0) and S2(0, k) = 0 # for n, k > 0
   S2(n, n) = 1
   S2(n + 1, k) = k * S2(n, k) + S2(n, k - 1)


Task
  • Write a routine (function, procedure, whatever) to find Stirling numbers of the second kind. There are several methods to generate Stirling numbers of the second kind. 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 Stirling numbers of the second kind, S2(n, k), up to S2(12, 12). it is optional to show the row / column for n == 0 and k == 0. It is optional to show places where S2(n, k) == 0 (when k > n).
  • If your language supports large integers, find and show here, on this page, the maximum value of S2(n, k) where n == 100.


See also


Related Tasks



ALGOL 68[edit]

Works with: ALGOL 68G version Any - tested with release 2.8.3.win32

Uses the LONG LONG INT mode of Algol 68g which allows large precision integers. As the default precision of LONG LONG INT is too small, the precision is specified via a pragmatic comment.

BEGIN
# show some Stirling numbers of the second kind #
 
# specify the precision of LONG LONG INT, somewhat under 160 digits are #
# needed for Stirling numbers of the second kind with n, k = 100 #
PR precision 160 PR
MODE SINT = LONG LONG INT;
 
# returns a triangular matrix of Stirling numbers up to max n, max n #
PROC make s2 = ( INT max n )REF[,]SINT:
BEGIN
REF[,]SINT s2 := HEAP[ 0 : max n, 0 : max n ]SINT;
FOR n FROM 0 TO max n DO FOR k FROM 0 TO max n DO s2[ n, k ] := 0 OD OD;
FOR n FROM 0 TO max n DO s2[ n, n ] := 1 OD;
FOR n FROM 0 TO max n - 1 DO
FOR k FROM 1 TO n DO
s2[ n + 1, k ] := k * s2[ n, k ] + s2[ n, k - 1 ];
OD
OD;
s2
END # make s2 # ;
# task requirements: #
# print Stirling numbers up to n, k = 12 #
BEGIN
INT max stirling = 12;
REF[,]SINT s2 = make s2( max stirling );
print( ( "Stirling numbers of the second kind:", newline ) );
print( ( " k" ) );
FOR k FROM 0 TO max stirling DO print( ( whole( k, -10 ) ) ) OD;
print( ( newline, " n", newline ) );
FOR n FROM 0 TO max stirling DO
print( ( whole( n, -2 ) ) );
FOR k FROM 0 TO n DO
print( ( whole( s2[ n, k ], -10 ) ) )
OD;
print( ( newline ) )
OD
END;
# find the maximum Stirling number with n = 100 #
BEGIN
INT max stirling = 100;
REF[,]SINT s2 = make s2( max stirling );
SINT max 100 := 0;
FOR k FROM 0 TO max stirling DO
IF s2[ max stirling, k ] > max 100 THEN max 100 := s2[ max stirling, k ] FI
OD;
print( ( "Maximum Stirling number of the second kind with n = 100:", newline ) );
print( ( whole( max 100, 0 ), newline ) )
END
END
Output:
Stirling numbers of the second kind:
 k         0         1         2         3         4         5         6         7         8         9        10        11        12
 n
 0         1
 1         0         1
 2         0         1         1
 3         0         1         3         1
 4         0         1         7         6         1
 5         0         1        15        25        10         1
 6         0         1        31        90        65        15         1
 7         0         1        63       301       350       140        21         1
 8         0         1       127       966      1701      1050       266        28         1
 9         0         1       255      3025      7770      6951      2646       462        36         1
10         0         1       511      9330     34105     42525     22827      5880       750        45         1
11         0         1      1023     28501    145750    246730    179487     63987     11880      1155        55         1
12         0         1      2047     86526    611501   1379400   1323652    627396    159027     22275      1705        66         1
Maximum Stirling number of the second kind with n = 100:
7769730053598745155212806612787584787397878128370115840974992570102386086289805848025074822404843545178960761551674

Factor[edit]

Works with: Factor version 0.99 development version 2019-07-10
USING: combinators.short-circuit formatting io kernel math
math.extras prettyprint sequences ;
RENAME: stirling math.extras => (stirling)
IN: rosetta-code.stirling-second
 
! Tweak Factor's in-built stirling function for k=0
: stirling ( n k -- m )
2dup { [ = not ] [ nip zero? ] } 2&&
[ 2drop 0 ] [ (stirling) ] if ;
 
"Stirling numbers of the second kind: n k stirling:" print
"n\\k" write 13 dup [ "%8d" printf ] each-integer nl
 
<iota> [
dup dup "%-2d " printf [0,b] [
stirling "%8d" printf
] with each nl
] each nl
 
"Maximum value from the 100 _ stirling row:" print
100 <iota> [ 100 swap stirling ] map supremum .
Output:
Stirling numbers of the second kind: n k stirling:
n\k       0       1       2       3       4       5       6       7       8       9      10      11      12
0         1
1         0       1
2         0       1       1
3         0       1       3       1
4         0       1       7       6       1
5         0       1      15      25      10       1
6         0       1      31      90      65      15       1
7         0       1      63     301     350     140      21       1
8         0       1     127     966    1701    1050     266      28       1
9         0       1     255    3025    7770    6951    2646     462      36       1
10        0       1     511    9330   34105   42525   22827    5880     750      45       1
11        0       1    1023   28501  145750  246730  179487   63987   11880    1155      55       1
12        0       1    2047   86526  611501 1379400 1323652  627396  159027   22275    1705      66       1

Maximum value from the 100 _ stirling row:
7769730053598745155212806612787584787397878128370115840974992570102386086289805848025074822404843545178960761551674

Go[edit]

package main
 
import (
"fmt"
"math/big"
)
 
func main() {
limit := 100
last := 12
s2 := make([][]*big.Int, limit+1)
for n := 0; n <= limit; n++ {
s2[n] = make([]*big.Int, limit+1)
for k := 0; k <= limit; k++ {
s2[n][k] = new(big.Int)
}
s2[n][n].SetInt64(int64(1))
}
var t big.Int
for n := 1; n <= limit; n++ {
for k := 1; k <= n; k++ {
t.SetInt64(int64(k))
t.Mul(&t, s2[n-1][k])
s2[n][k].Add(&t, s2[n-1][k-1])
}
}
fmt.Println("Stirling numbers of the second kind: S2(n, k):")
fmt.Printf("n/k")
for i := 0; i <= last; i++ {
fmt.Printf("%9d ", 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("%9d ", s2[n][k])
}
fmt.Println()
}
fmt.Println("\nMaximum value from the S2(100, *) row:")
max := new(big.Int).Set(s2[limit][0])
for k := 1; k <= limit; k++ {
if s2[limit][k].Cmp(max) > 0 {
max.Set(s2[limit][k])
}
}
fmt.Println(max)
fmt.Printf("which has %d digits.\n", len(max.String()))
}
Output:
Stirling numbers of the second kind: S2(n, k):
n/k        0         1         2         3         4         5         6         7         8         9        10        11        12 
------------------------------------------------------------------------------------------------------------------------------------
 0         1 
 1         0         1 
 2         0         1         1 
 3         0         1         3         1 
 4         0         1         7         6         1 
 5         0         1        15        25        10         1 
 6         0         1        31        90        65        15         1 
 7         0         1        63       301       350       140        21         1 
 8         0         1       127       966      1701      1050       266        28         1 
 9         0         1       255      3025      7770      6951      2646       462        36         1 
10         0         1       511      9330     34105     42525     22827      5880       750        45         1 
11         0         1      1023     28501    145750    246730    179487     63987     11880      1155        55         1 
12         0         1      2047     86526    611501   1379400   1323652    627396    159027     22275      1705        66         1 

Maximum value from the S2(100, *) row:
7769730053598745155212806612787584787397878128370115840974992570102386086289805848025074822404843545178960761551674
which has 115 digits.

Julia[edit]

using Combinatorics
 
const s2cache = Dict()
 
function stirlings2(n, k)
if haskey(s2cache, Pair(n, k))
return s2cache[Pair(n, k)]
elseif n < 0
throw(DomainError(n, "n must be nonnegative"))
elseif n == k == 0
return one(n)
elseif n == 0 || k == 0
return zero(n)
elseif k == n - 1
return binomial(n, 2)
elseif k == 2
return 2^(n-1) - 1
end
 
ret = k * stirlings2(n - 1, k) + stirlings2(n - 1, k - 1)
s2cache[Pair(n, k)] = ret
return ret
 
end
 
function printstirling2table(kmax)
println(" ", mapreduce(i -> lpad(i, 10), *, 0:kmax))
 
sstring(n, k) = begin i = stirlings2(n, k); lpad(k > n && i == 0 ? "" : i, 10) end
 
for n in 0:kmax
println(rpad(n, 2) * mapreduce(k -> sstring(n, k), *, 0:kmax))
end
end
 
printstirling2table(12)
println("\nThe maximum for stirling2(100, _) is: ", maximum(k-> stirlings2(BigInt(100), BigInt(k)), 1:100))
 
 
Output:
           0         1         2         3         4         5         6         7         8         9        10        11        12
0          1
1          0         1
2          0         1         1
3          0         1         3         1
4          0         1         7         6         1
5          0         1        15        25        10         1
6          0         1        31        90        65        15         1
7          0         1        63       301       350       140        21         1
8          0         1       127       966      1701      1050       266        28         1
9          0         1       255      3025      7770      6951      2646       462        36         1
10         0         1       511      9330     34105     42525     22827      5880       750        45         1
11         0         1      1023     28501    145750    246730    179487     63987     11880      1155        55         1
12         0         1      2047     86526    611501   1379400   1323652    627396    159027     22275      1705        66         1

The maximum for stirling2(100, _) is:
7769730053598745155212806612787584787397878128370115840974992570102386086289805848025074822404843545178960761551674

Perl[edit]

use strict;
use warnings;
use bigint;
use feature 'say';
use feature 'state';
no warnings 'recursion';
use List::Util qw(max);
 
sub Stirling2 {
my($n, $k) = @_;
my $n1 = $n - 1;
return 1 if $n1 == $k;
return 0 unless $n1 && $k;
state %seen;
return ($seen{"{$n1}|{$k}" } //= Stirling2($n1,$k ) * $k) +
($seen{"{$n1}|{$k-1}"} //= Stirling2($n1,$k-1))
}
 
my $upto = 12;
my $width = 1 + length max map { Stirling2($upto+1,$_) } 0..$upto+1;
 
say 'Unsigned Stirling2 numbers of the second kind: S2(n, k):';
print 'n\k' . sprintf "%${width}s"x(1+$upto)."\n", 0..$upto;
 
for my $row (1..$upto+1) {
printf '%-3d', $row-1;
printf "%${width}d", Stirling2($row, $_) for 0..$row-1;
print "\n";
}
 
say "\nMaximum value from the S2(100, *) row:";
say max map { Stirling2(101,$_) } 0..100;
Output:
Unsigned Stirling2 numbers of the second kind: S2(n, k):
n\k       0       1       2       3       4       5       6       7       8       9      10      11      12
0         1
1         0       1
2         0       1       1
3         0       1       3       1
4         0       1       7       6       1
5         0       1      15      25      10       1
6         0       1      31      90      65      15       1
7         0       1      63     301     350     140      21       1
8         0       1     127     966    1701    1050     266      28       1
9         0       1     255    3025    7770    6951    2646     462      36       1
10        0       1     511    9330   34105   42525   22827    5880     750      45       1
11        0       1    1023   28501  145750  246730  179487   63987   11880    1155      55       1
12        0       1    2047   86526  611501 1379400 1323652  627396  159027   22275    1705      66       1

Maximum value from the S2(100, *) row:
7769730053598745155212806612787584787397878128370115840974992570102386086289805848025074822404843545178960761551674

Perl 6[edit]

Works with: Rakudo version 2019.07.1
sub Stirling2 (Int \n, Int \k) {
((1,), { (0, |@^last) »+« (|(@^last »*« @^last.keys), 0) }*)[n;k]
}
 
my $upto = 12;
 
my $mx = (1..^$upto).map( { Stirling2($upto, $_) } ).max.chars;
 
put 'Stirling numbers of the second kind: S2(n, k):';
put 'n\k', (0..$upto)».fmt: "%{$mx}d";
 
for 0..$upto -> $row {
$row.fmt('%-3d').print;
put (0..$row).map( { Stirling2($row, $_) } )».fmt: "%{$mx}d";
}
 
say "\nMaximum value from the S2(100, *) row:";
say (^100).map( { Stirling2 100, $_ } ).max;
Output:
Stirling numbers of the second kind: S2(n, k):
n\k      0       1       2       3       4       5       6       7       8       9      10      11      12
0        1
1        0       1
2        0       1       1
3        0       1       3       1
4        0       1       7       6       1
5        0       1      15      25      10       1
6        0       1      31      90      65      15       1
7        0       1      63     301     350     140      21       1
8        0       1     127     966    1701    1050     266      28       1
9        0       1     255    3025    7770    6951    2646     462      36       1
10       0       1     511    9330   34105   42525   22827    5880     750      45       1
11       0       1    1023   28501  145750  246730  179487   63987   11880    1155      55       1
12       0       1    2047   86526  611501 1379400 1323652  627396  159027   22275    1705      66       1

Maximum value from the S2(100, *) row:
7769730053598745155212806612787584787397878128370115840974992570102386086289805848025074822404843545178960761551674

Phix[edit]

Library: mpfr
Translation of: Go
include mpfr.e
 
constant lim = 100,
lim1 = lim+1,
last = 12
sequence s2 = repeat(0,lim1)
for n=1 to lim1 do
s2[n] = mpz_inits(lim1)
mpz_set_si(s2[n][n],1)
end for
mpz {t, m100} = mpz_inits(2)
for n=1 to lim do
for k=1 to n do
mpz_set_si(t,k)
mpz_mul(t,t,s2[n][k+1])
mpz_add(s2[n+1][k+1],t,s2[n][k])
end for
end for
printf(1,"Stirling numbers of the second kind: S2(n, k):\n n k:")
for i=0 to last do
printf(1,"%5d ", i)
end for
printf(1,"\n---  %s\n",repeat('-',last*10+5))
for n=0 to last do
printf(1,"%2d ", n)
for k=1 to n+1 do
mpfr_printf(1,"%9Zd ", s2[n+1][k])
end for
printf(1,"\n")
end for
for k=1 to lim1 do
mpz s100k = s2[lim1][k]
if mpz_cmp(s100k,m100) > 0 then
mpz_set(m100,s100k)
end if
end for
printf(1,"\nThe maximum S2(100,k): %s\n",shorten(mpz_get_str(m100)))
Output:
Stirling numbers of the second kind: S2(n, k):
 n  k:     0         1         2         3         4         5         6         7         8         9        10        11        12
---   ------------------------------------------------------------------------------------------------------------------------------
 0         1
 1         0         1
 2         0         1         1
 3         0         1         3         1
 4         0         1         7         6         1
 5         0         1        15        25        10         1
 6         0         1        31        90        65        15         1
 7         0         1        63       301       350       140        21         1
 8         0         1       127       966      1701      1050       266        28         1
 9         0         1       255      3025      7770      6951      2646       462        36         1
10         0         1       511      9330     34105     42525     22827      5880       750        45         1
11         0         1      1023     28501    145750    246730    179487     63987     11880      1155        55         1
12         0         1      2047     86526    611501   1379400   1323652    627396    159027     22275      1705        66         1

Maximum value from the S2(100, *) row: 7769730053598745155...3545178960761551674 (115 digits)

REXX[edit]

Some extra code was added to minimize the displaying of the column widths.

/*REXX program to compute and display   Stirling numbers   of the second kind.          */
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, 2*lim) /*(over) specify maximum number in grid*/
@.=
do j=0 for lim+1
@.j.j = 1; if j==0 then iterate /*define the right descending diagonal.*/
@.0.j = 0; @.j.0 = 0 /* " " zero values. */
end /*j*/
max#.= 0 /* [↓] calculate values for the grid. */
do n=0 for lim+1; np= n + 1
do k=1 for lim; km= k - 1
@.np.k = k * @.n.k + @.n.km /*calculate a number in the grid. */
max#.k= max(max#.k, @.n.k) /*find the maximum value for a column. */
max#.b= max(max#.b, @.n.k) /*find the maximum value for all rows. */
end /*k*/
end /*n*/
/* [↓] only show the maximum value ? */
do k=0 for lim+1 /*find max column width for each column*/
max#.a= max#.a + length(max#.k)
end /*k*/
 
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
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*/ /*stick a fork in it, we're all done. */
output   when using the default input:
row ══════════════════════════════columns══════════════════════════════
  0 1
  1 0 1
  2 0 1    1
  3 0 1    3     1
  4 0 1    7     6      1
  5 0 1   15    25     10       1
  6 0 1   31    90     65      15       1
  7 0 1   63   301    350     140      21      1
  8 0 1  127   966   1701    1050     266     28      1
  9 0 1  255  3025   7770    6951    2646    462     36     1
 10 0 1  511  9330  34105   42525   22827   5880    750    45    1
 11 0 1 1023 28501 145750  246730  179487  63987  11880  1155   55  1
 12 0 1 2047 86526 611501 1379400 1323652 627396 159027 22275 1705 66 1
output   when using the input of:     -100
The maximum value  (which has  115  decimal digits):
7769730053598745155212806612787584787397878128370115840974992570102386086289805848025074822404843545178960761551674

Sidef[edit]

func S2(n, k) {     # Stirling numbers of the second kind
stirling2(n, k)
}
 
const r = (0..12)
 
var triangle = r.map {|n| 0..n -> map {|k| S2(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 S2(#{n}, *) row:"
say { S2(n, _) }.map(^n).max
}
Output:
n\k 0 1    2     3      4       5       6      7      8     9   10 11 12
0   1
1   0 1
2   0 1    1
3   0 1    3     1
4   0 1    7     6      1
5   0 1   15    25     10       1
6   0 1   31    90     65      15       1
7   0 1   63   301    350     140      21      1
8   0 1  127   966   1701    1050     266     28      1
9   0 1  255  3025   7770    6951    2646    462     36     1
10  0 1  511  9330  34105   42525   22827   5880    750    45    1
11  0 1 1023 28501 145750  246730  179487  63987  11880  1155   55  1
12  0 1 2047 86526 611501 1379400 1323652 627396 159027 22275 1705 66 1

Maximum value from the S2(100, *) row:
7769730053598745155212806612787584787397878128370115840974992570102386086289805848025074822404843545178960761551674

Alternatively, the S2(n,k) function can be defined as:

func S2((0), (0)) { 1 }
func S2(_, (0)) { 0 }
func S2((0), _) { 0 }
func S2(n, k) is cached { S2(n-1, k)*k + S2(n-1, k-1) }

zkl[edit]

fcn stirling2(n,k){
var seen=Dictionary(); // cache for recursion
if(n==k) return(1); // (0.0)==1
if(n<1 or k<1) return(0);
z1,z2 := "%d,%d".fmt(n-1,k), "%d,%d".fmt(n-1,k-1);
if(Void==(s1 := seen.find(z1))){ s1 = seen[z1] = stirling2(n-1,k) }
if(Void==(s2 := seen.find(z2))){ s2 = seen[z2] = stirling2(n-1,k-1) }
k*s1 + s2; // k is first to cast to BigInt (if using BigInts)
}
// calculate entire table (cached), find max, find num digits in max
N,mx := 12, [1..N].apply(fcn(n){ [1..n].apply(stirling2.fp(n)) }).flatten() : (0).max(_);
fmt:="%%%dd".fmt("%d".fmt(mx.numDigits + 1)).fmt; // "%9d".fmt
println("Stirling numbers of the second kind: S2(n,k):");
println("n\\k",[0..N].pump(String,fmt));
foreach row in ([0..N]){
println("%3d".fmt(row), [0..row].pump(String, stirling2.fp(row), fmt));
}
Output:
Stirling numbers of the second kind: S2(n,k):
n\k       0       1       2       3       4       5       6       7       8       9      10      11      12
  0       1
  1       0       1
  2       0       1       1
  3       0       1       3       1
  4       0       1       7       6       1
  5       0       1      15      25      10       1
  6       0       1      31      90      65      15       1
  7       0       1      63     301     350     140      21       1
  8       0       1     127     966    1701    1050     266      28       1
  9       0       1     255    3025    7770    6951    2646     462      36       1
 10       0       1     511    9330   34105   42525   22827    5880     750      45       1
 11       0       1    1023   28501  145750  246730  179487   63987   11880    1155      55       1
 12       0       1    2047   86526  611501 1379400 1323652  627396  159027   22275    1705      66       1
Library: GMP
GNU Multiple Precision Arithmetic Library
var [const] BI=Import("zklBigNum");  // libGMP
N=100;
S2100:=[BI(2)..N].apply(stirling2.fp(BI(N))).reduce(fcn(m,n){ m.max(n) });
println("Maximum value from the S2(%d,*) row (%d digits):".fmt(N,S2100.numDigits));
println(S2100);
Output:
Maximum value from the S2(100,*) row (115 digits):
7769730053598745155212806612787584787397878128370115840974992570102386086289805848025074822404843545178960761551674