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

1, 1, 1, 3, 8, 22, 75, 267, 1119, 4965, 22694, 117090, 670621, 3866503, 24113829, 161085223, 1120025702, 8121648620, 62083083115, 492273775141, 4074919882483

Offset: 0

Author

Gus Wiseman, Apr 04 2023

Keywords

• nonn
• more

Examples

			The a(1) = 1 through a(4) = 8 set partitions:
  {{1}}  {{1}{2}}  {{123}}      {{1}{234}}
                   {{13}{2}}    {{12}{34}}
                   {{1}{2}{3}}  {{123}{4}}
                                {{13}{24}}
                                {{14}{23}}
                                {{1}{24}{3}}
                                {{13}{2}{4}}
                                {{1}{2}{3}{4}}
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).
		

Crossrefs

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

For median instead of mean we have A361911.

A000110 counts set partitions.

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

A308037 counts set partitions with integer mean block-size.

A327475 counts subsets with integer mean, median A000975.

A327481 counts subsets by mean, median A013580.

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

Cf. A067659, A326515, A326516, A326521.

Programs

Mathematica

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
Table[Length[Select[sps[Range[n]],IntegerQ[Total[Mean/@#]]&]],{n,6}]

Extensions

a(14)-a(20) from Christian Sievers, May 12 2025