A191716 a(n,k) equals (1/n!) multiplied by the count of permutations with cycle length k in all products u v u^-1 v^-1 over all permutations u and v of length n.

1, 0, 2, 3, 0, 3, 0, 19, 0, 5, 40, 0, 73, 0, 7, 0, 492, 0, 217, 0, 11, 1260, 0, 3225, 0, 540, 0, 15, 0, 24096, 0, 14968, 0, 1234, 0, 22, 72576, 0, 232156, 0, 55594, 0, 2524, 0, 30, 0, 1922148, 0, 1524823, 0, 176800, 0, 4987, 0, 42, 6652800, 0, 24999984, 0, 7758160, 0, 496680, 0, 9120, 0, 56, 0, 227963280, 0, 216975032, 0, 32769481, 0, 1277331, 0, 16399, 0, 77

Offset

1,3

Comments

Row sums equal n! by definition.

Links

Table of n, a(n) for n=1..78.

R. Stanley, Hook Lengths and Contents slides 26 & 27

Example

1;
0, 2;
3, 0, 3;
0, 19, 0, 5;
40, 0, 73, 0, 7;
0, 492, 0, 217, 0, 11;
1260, 0, 3225, 0, 540, 0, 15;
0, 24096, 0, 14968, 0, 1234, 0, 22;

Mathematica

(*slow:*)
Table[Rest@CoefficientList[Apply[Plus, Flatten[Outer[ q^Length[ ToCycles[#1[[#2]][[InversePermutation[#1]]][[InversePermutation[#2]]]]] &, Permutations[w], Permutations[w], 1]]], q]/w!, {w, 4}]//Expand;
(*fast:*)
content[(p_)?PartitionQ]:= Block[{le= Max[p], ferr =(PadLeft[1+ 0*Range[#1], Max[p]]&) /@ p}, DeleteCases[ MapIndexed[-le+ Range[le, 1, -1]- #1- Tr[#2]&, 0*ferr]*ferr, 0, -1]+ le];
Table[Rest@ CoefficientList[ Apply[Plus, Apply[Times, q + Flatten[content[#]]] & /@ Partitions[ k ]] , q], {k, 12}]

Crossrefs

Cf. A191714

Sequence in context: A128621 A132385 A358840 * A089235 A347794 A317948

Adjacent sequences: A191713 A191714 A191715 * A191717 A191718 A191719

Keyword

nonn,tabl

Author

Wouter Meeussen, Jun 12 2011

Status

approved