cp's OEIS Frontend

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.

%I A191716 #8 Mar 30 2012 18:37:45
%S A191716 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,
%T A191716 0,15,0,24096,0,14968,0,1234,0,22,72576,0,232156,0,55594,0,2524,0,30,
%U A191716 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
%N 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.
%C A191716 Row sums equal n! by definition.
%H A191716 R. Stanley, <a href="http://www-math.mit.edu/~rstan/transparencies/hooks.pdf">Hook Lengths and Contents slides 26 & 27</a>
%e A191716 1;
%e A191716 0, 2;
%e A191716 3, 0, 3;
%e A191716 0, 19, 0, 5;
%e A191716 40, 0, 73, 0, 7;
%e A191716 0, 492, 0, 217, 0, 11;
%e A191716 1260, 0, 3225, 0, 540, 0, 15;
%e A191716 0, 24096, 0, 14968, 0, 1234, 0, 22;
%t A191716 (*slow:*)
%t A191716 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;
%t A191716 (*fast:*)
%t A191716 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];
%t A191716 Table[Rest@ CoefficientList[ Apply[Plus, Apply[Times, q + Flatten[content[#]]] & /@ Partitions[ k ]] , q], {k, 12}]
%Y A191716 Cf. A191714
%K A191716 nonn,tabl
%O A191716 1,3
%A A191716 _Wouter Meeussen_, Jun 12 2011