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A023997 Number of block permutations on an n-set. 8
1, 1, 3, 25, 339, 6721, 179643, 6166105, 262308819, 13471274401, 818288740923, 57836113793305, 4693153430067699, 432360767273547841, 44794795522199781243, 5176959027946049635225, 662704551840482536170579 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
A block permutation of a set X is a bijection between two quotient sets of X (of necessarily equal rank).
Number of labeled partitions of (n,n) into pairs (i,j) where there are n black objects labeled 1..n and n white objects labeled 1..n. Each partition must have at least one black object and at least one white object. - Christian G. Bower, Jun 03 2005
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..288 (terms 0..45 from Vincenzo Librandi)
Zhanar Berikkyzy, Pamela E. Harris, Anna Pun, Catherine Yan, and Chenchen Zhao, Combinatorial Identities for Vacillating Tableaux, arXiv:2308.14183 [math.CO], 2023. See p. 22.
D. G. FitzGerald and Jonathan Leech, Dual symmetric inverse monoids and representation theory, J. Australian Mathematical Society (Series A), Vol. 64 (1998), pp. 345-367.
FORMULA
a(0)=1, a(n) = Sum_{k=1..n} k! * S2(n,k)^2, S2(n,k) are the Stirling numbers of the second kind.
EXAMPLE
For n=3, there are the 3! ordinary permutations (of rank 3), 18 block permutations of rank 2 (2! for each pair of partitions of rank 2) and the single rank 1 one.
MATHEMATICA
Table[Sum[StirlingS2[n, k]^2k!, {k, 0, n}], {n, 0, 100}] (* Emanuele Munarini, Jul 04 2011 *)
PROG
(Maxima) makelist(sum(stirling2(n, k)^2*k!, k, 0, n), n, 0, 24); /* Emanuele Munarini, Jul 04 2011 */
(PARI) a(n) = if (n==0, 1, sum(k=1, n, k!*stirling(n, k, 2)^2)); \\ Michel Marcus, Jun 18 2019
CROSSREFS
Sequence in context: A181085 A143635 A246756 * A154961 A322760 A325286
KEYWORD
easy,nonn,nice
AUTHOR
Des FitzGerald (D.FitzGerald(AT)utas.edu.au)
EXTENSIONS
More terms from Christian G. Bower, Jun 03 2005
STATUS
approved

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Last modified July 13 02:50 EDT 2024. Contains 374265 sequences. (Running on oeis4.)