Stirling numbers of the second kind: Difference between revisions
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Revision as of 20:53, 15 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
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