OFFSET
0,3
COMMENTS
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.
EXAMPLE
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) ).
MATHEMATICA
Needs["Combinatorica`"]; f[partition_] := Normal[SparseArray[ Level[Map[Tuples[#, 2] &, partition], {2}] -> 1]]; Table[er = Map[f, SetPartitions[n]]; Length[Level[
Table[Select[er, Clip[er[[i]].#] == Clip[#.er[[i]]] &], {i, 1, Length[er]}], {2}]], {n, 0, 8}]
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Geoffrey Critzer, May 30 2022
EXTENSIONS
a(9) from Vaclav Kotesovec, May 31 2022
STATUS
approved