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a(n) = A322437(A002110(n)).
1

%I #12 Dec 11 2020 11:36:57

%S 0,0,0,0,3,30,315,4830,96453,2296350

%N a(n) = A322437(A002110(n)).

%C For n > 0, a(n) gives the number of unordered pairs of set partitions of {1,...,n} where no block of the other is a subset (or equal) to any block of the other. See A322441.

%F For n > 0, a(n) = A322441(n)/2.

%e The a(4) = 3 such (unordered) pairs of set partitions of {1,2,3,4} are:

%e {{1,2},{3,4}}|{{1,3},{2,4}}

%e {{1,2},{3,4}}|{{1,4},{2,3}}

%e {{1,3},{2,4}}|{{1,4},{2,3}}.

%t Block[{f}, f[n_] := If[n <= 1, {{}}, Join @@ Table[Map[Prepend[#, d] &, Select[f[n/d], Min @@ # >= d &]], {d, Rest[Divisors[n]]}]]; Map[Length[Select[Subsets[f[#], {2}], And[! Or @@ Divisible @@@ #, ! Or @@ Divisible @@@ Reverse /@ #] &@ Tuples[#] &]] &, FoldList[Times, 1, Prime@ Range@ 7]] ] (* _Michael De Vlieger_, Dec 10 2020, after _Gus Wiseman_ at A322437 *)

%Y Cf. A002110, A322437, A322441.

%K nonn,more

%O 0,5

%A _Antti Karttunen_, Dec 10 2020