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

Offset

0,4

Links

Table of n, a(n) for n=0..13.

Example

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).

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}]

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.

Sequence in context: A369791 A189944 A148771 * A148772 A148773 A148774

Adjacent sequences: A361863 A361864 A361865 * A361867 A361868 A361869

Keyword

nonn,more

Author

Gus Wiseman, Apr 04 2023

Status

approved