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A349739 Number of ordered pairs of commuting elements (partial permutations) in the symmetric inverse semigroup on [n].

%I A349739 #43 Dec 20 2021 18:15:51
%S A349739 1,4,31,310,3925,58936,1032979,20600266,461742985
%N A349739 Number of ordered pairs of commuting elements (partial permutations) in the symmetric inverse semigroup on [n].
%D A349739 Stephen Lipscomb, Symmetric Inverse Semigroups, Mathematical Surveys and Monographs, Volume 46, 1996, Chapters 3,4,5.
%t A349739 x[list_] := If[list == {}, 1, Apply[Times, list] Apply[Times,  Table[Count[list, i]!, {i, 1, Max[list]}]]]; y[list_] := If[list == {}, 1, Apply[Times,Table[Count[list, i]!, {i, 1, Max[list]}]]];
%t A349739 c[n_, pair_] := n!/(x[pair[[1]]] y[pair[[2]]]); n[k_, list_] := Count[list, k];
%t A349739 m[k_, list_] := Sum[Binomial[n[k, list], j]^2 j! k^j, {j, 0, n[k, list]}];
%t A349739 xp[list_] := Apply[Times, Table[m[k, list], {k, 1, Max[{1, list}]}]];
%t A349739 partialPermMatrices1[n_] := Module[{im = PadRight[IdentityMatrix[n], {n + 1, n}]},
%t A349739   Sort@Map[Extract[im, List /@ #] &]@ Permutations[Join[ConstantArray[n + 1, n], Range@n], {n}]]; s[list_] := Total[Map[Apply[Times, #] &,Map[Min, Map[list[[#]] &, Map[Position[#, 1] &, partialPermMatrices1[Length[list]]], {2}], {2}]]]; Table[(Map[s,Level[Table[Level[Table[Table[{IntegerPartitions[nn - k][[i]],     IntegerPartitions[k][[j]]}, {i, 1,PartitionsP[nn - k]}], {j, 1,PartitionsP[k]}], {2}], {k,0, nn}], {2}][[All, 2]]])*(Map[xp, Level[Table[ Level[Table[Table[{IntegerPartitions[nn - k][[i]], IntegerPartitions[k][[j]]}, {i, 1, PartitionsP[nn - k]}], {j, 1, PartitionsP[k]}], {2}], {k, 0, nn}], {2}][[All, 1]]])*(Map[c[nn, #] &,Level[Table[Level[Table[Table[{IntegerPartitions[nn - k][[i]],IntegerPartitions[k][[j]]}, {i, 1, PartitionsP[nn - k]}], {j, 1,PartitionsP[k]}], {2}], {k, 0, nn}], {2}]]) // Total, {nn, 0, 7}]
%Y A349739 Cf. A002720, A000712 (number of conjugacy classes).
%K A349739 nonn,more
%O A349739 0,2
%A A349739 _Geoffrey Critzer_, Dec 19 2021