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A361866 Number of set partitions of {1..n} with block-means summing to an integer.

%I A361866 #11 May 12 2025 19:06:48
%S A361866 1,1,1,3,8,22,75,267,1119,4965,22694,117090,670621,3866503,24113829,
%T A361866 161085223,1120025702,8121648620,62083083115,492273775141,
%U A361866 4074919882483
%N A361866 Number of set partitions of {1..n} with block-means summing to an integer.
%e A361866 The a(1) = 1 through a(4) = 8 set partitions:
%e A361866   {{1}}  {{1}{2}}  {{123}}      {{1}{234}}
%e A361866                    {{13}{2}}    {{12}{34}}
%e A361866                    {{1}{2}{3}}  {{123}{4}}
%e A361866                                 {{13}{24}}
%e A361866                                 {{14}{23}}
%e A361866                                 {{1}{24}{3}}
%e A361866                                 {{13}{2}{4}}
%e A361866                                 {{1}{2}{3}{4}}
%e A361866 The set partition y = {{1,2},{3,4}} has block-means {3/2,7/2}, with sum 5, so y is counted under a(4).
%t A361866 sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t A361866 Table[Length[Select[sps[Range[n]],IntegerQ[Total[Mean/@#]]&]],{n,6}]
%Y A361866 For mean instead of sum we have A361865, for median A361864.
%Y A361866 For median instead of mean we have A361911.
%Y A361866 A000110 counts set partitions.
%Y A361866 A067538 counts partitions with integer mean, ranks A326836, strict A102627.
%Y A361866 A308037 counts set partitions with integer mean block-size.
%Y A361866 A327475 counts subsets with integer mean, median A000975.
%Y A361866 A327481 counts subsets by mean, median A013580.
%Y A361866 Cf. A007837, A035470, A038041, A275714, A275780, A326512, A326513.
%Y A361866 Cf. A067659, A326515, A326516, A326521.
%K A361866 nonn,more
%O A361866 0,4
%A A361866 _Gus Wiseman_, Apr 04 2023
%E A361866 a(14)-a(20) from _Christian Sievers_, May 12 2025