A361865 Number of set partitions of {1..n} such that the mean of the means of the blocks is an integer.

1, 0, 3, 2, 12, 18, 101, 232, 1547, 3768, 24974, 116728

Offset

1,3

Links

Table of n, a(n) for n=1..12.

Example

The set partition y = {{1,4},{2,5},{3}} has block-means {5/2,7/2,3}, with mean 3, so y is counted under a(5).
The a(1) = 1 through a(5) = 12 set partitions:
  {{1}}  .  {{123}}      {{1}{234}}  {{12345}}
            {{13}{2}}    {{123}{4}}  {{1245}{3}}
            {{1}{2}{3}}              {{135}{24}}
                                     {{15}{234}}
                                     {{1}{234}{5}}
                                     {{12}{3}{45}}
                                     {{135}{2}{4}}
                                     {{14}{25}{3}}
                                     {{15}{24}{3}}
                                     {{1}{24}{3}{5}}
                                     {{15}{2}{3}{4}}
                                     {{1}{2}{3}{4}{5}}

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[Mean[Mean/@#]]&]], {n, 6}]

Crossrefs

For median instead of mean we have A361864.

For sum instead of outer mean we have A361866, median A361911.

A000110 counts set partitions.

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

A308037 appears to count set partitions whose block-sizes have integer mean.

A327475 counts subsets with integer mean, median A000975.

Cf. A007837, A035470, A038041, A275714, A275780, A326512, A326513, A326521, A326537, A327481.

Sequence in context: A113205 A136657 A006774 * A356857 A086551 A254215

Adjacent sequences: A361862 A361863 A361864 * A361866 A361867 A361868

Keyword

nonn,more

Author

Gus Wiseman, Apr 04 2023

Status

approved