

A023997


Number of block permutations on an nset.


8



1, 1, 3, 25, 339, 6721, 179643, 6166105, 262308819, 13471274401, 818288740923, 57836113793305, 4693153430067699, 432360767273547841, 44794795522199781243, 5176959027946049635225, 662704551840482536170579
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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)
D. G. FitzGerald and Jonathan Leech, Dual symmetric inverse monoids and representation theory, J. Australian Mathematical Society (Series A), Vol. 64 (1998), pp. 345367.


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

Cf. A023998, A002720, A014235, A111420.
Sequence in context: A181085 A143635 A246756 * A154961 A322760 A325286
Adjacent sequences: A023994 A023995 A023996 * A023998 A023999 A024000


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



