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

Author

Wouter Meeussen, Jun 12 2011

Keywords

• nonn
• tabl

Comments

Row sums equal n! by definition.

Examples

			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;
		

Links

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

Crossrefs

Cf. A191714

Programs

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}]