A023997 Number of block permutations on an n-set.

1, 1, 3, 25, 339, 6721, 179643, 6166105, 262308819, 13471274401, 818288740923, 57836113793305, 4693153430067699, 432360767273547841, 44794795522199781243, 5176959027946049635225, 662704551840482536170579

Offset: 0

Author

Des FitzGerald (D.FitzGerald(AT)utas.edu.au)

Keywords

• easy
• nonn
• nice

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

Examples

			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.
		

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.

Crossrefs

Cf. A023998, A002720, A014235, A111420.

Programs

Mathematica

Table[Sum[StirlingS2[n,k]^2k!,{k,0,n}],{n,0,100}] (* Emanuele Munarini, Jul 04 2011 *)

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

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.

Extensions

More terms from Christian G. Bower, Jun 03 2005