A354458 Number of commuting pairs of equivalence relations on [n].

1, 1, 4, 19, 117, 864, 7459, 73749, 818960, 10078023

Offset: 0

Author

Geoffrey Critzer, May 30 2022

Keywords

• nonn
• more

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.

Examples

			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) ).
		

Crossrefs

Cf. A001247, A000110.

Programs

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}]

Extensions

a(9) from Vaclav Kotesovec, May 31 2022