%I #8 Apr 07 2023 09:27:59

%S 1,1,1,3,8,22,75,267,1119,4965,22694,117090,670621,3866503

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

%e The a(1) = 1 through a(4) = 8 set partitions:

%e {{1}} {{1}{2}} {{123}} {{1}{234}}

%e {{13}{2}} {{12}{34}}

%e {{1}{2}{3}} {{123}{4}}

%e {{13}{24}}

%e {{14}{23}}

%e {{1}{24}{3}}

%e {{13}{2}{4}}

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

%e 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 sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];

%t Table[Length[Select[sps[Range[n]],IntegerQ[Total[Mean/@#]]&]],{n,6}]

%Y For mean instead of sum we have A361865, for median A361864.

%Y For median instead of mean we have A361911.

%Y A000110 counts set partitions.

%Y A067538 counts partitions with integer mean, ranks A326836, strict A102627.

%Y A308037 counts set partitions with integer mean block-size.

%Y A327475 counts subsets with integer mean, median A000975.

%Y A327481 counts subsets by mean, median A013580.

%Y Cf. A007837, A035470, A038041, A275714, A275780, A326512, A326513.

%Y Cf. A067659, A326515, A326516, A326521.

%K nonn,more

%O 0,4

%A _Gus Wiseman_, Apr 04 2023