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%I #12 Jun 17 2022 15:56:31
%S 1,1,4,19,117,864,7459,73749,818960,10078023
%N Number of commuting pairs of equivalence relations on [n].
%C More precisely, a(n) is the number of ordered pairs (S,T) of equivalence relations on [n] such that S*T=T*S where the operation * is composition of relations. The composition of equivalence relations is not generally an equivalence relation. S*T=T*S if and only if S*T is the smallest equivalence relation that contains both S and T.
%e Let S = 1/24/3 and T = 13/2/4 be equivalence relations on [4]. Then S*T = T*S = 13/24 so (S,T) is an example of a commuting pair of equivalence relations (as well as (T,S) ).
%t Needs["Combinatorica`"];f[partition_] := Normal[SparseArray[ Level[Map[Tuples[#, 2] &, partition], {2}] -> 1]]; Table[er = Map[f,SetPartitions[n]]; Length[Level[
%t Table[Select[er, Clip[er[[i]].#] == Clip[#.er[[i]]] &], {i, 1,Length[er]}], {2}]], {n, 0, 8}]
%Y Cf. A001247, A000110.
%K nonn,more
%O 0,3
%A _Geoffrey Critzer_, May 30 2022
%E a(9) from _Vaclav Kotesovec_, May 31 2022
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