Stirling numbers of the second kind: Difference between revisions
(→{{header|REXX}}: added the REXX computer programming language for this task.) |
m (→{{header|PEXX}}: changed a comment.) |
||
| Line 145: | Line 145: | ||
say max# /*display maximum number in the grid. */ |
say max# /*display maximum number in the grid. */ |
||
exit /*stick a fork in it, we're all done. */ |
exit /*stick a fork in it, we're all done. */ |
||
end /*[↑] the 100th row is when LIM is 99 |
end /* [↑] the 100th row is when LIM is 99*/ |
||
wi= max(3, length(lim+1) ) /*the maximum width of the grid's index*/ |
wi= max(3, length(lim+1) ) /*the maximum width of the grid's index*/ |
||
say 'row' center('columns', (w+1)*(lim+1), '═') /*display the header of the grid. */ |
say 'row' center('columns', (w+1)*(lim+1), '═') /*display the header of the grid. */ |
||
Revision as of 05:35, 16 August 2019
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 Sterling numbers of the second kind. There are several methods to generate Sterling 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
Factor
<lang factor>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 .</lang>
- 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
Perl 6
<lang perl6>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;</lang>
- 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
PEXX
<lang>/*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#= max(max#, @.np.k) /*find " " value " " " */
end /*k*/
end /*n*/
/* [↓] only show the maximum value ? */
w= length(max#) /*calculate max width of all numbers. */ if olim<0 then do; say 'The maximum value (which has ' w " decimal digits):"
say max# /*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', (w+1)*(lim+1), '═') /*display the 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, w) /*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. */</lang>
- 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: -99
The maximum value (which has 115 decimal digits): 7769730053598745155212806612787584787397878128370115840974992570102386086289805848025074822404843545178960761551674