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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. 0
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 (list; table; graph; refs; listen; history; text; internal format)
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: A241070 A128621 A132385 * A089235 A317948 A334291

Adjacent sequences:  A191713 A191714 A191715 * A191717 A191718 A191719

KEYWORD

nonn,tabl

AUTHOR

Wouter Meeussen, Jun 12 2011

STATUS

approved

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Last modified April 23 10:15 EDT 2021. Contains 343204 sequences. (Running on oeis4.)