A361910 Number of set partitions of {1..n} such that the mean of the means of the blocks is (n+1)/2.

1, 2, 3, 7, 12, 47, 99, 430, 1379, 5613, 21416, 127303

Offset

1,2

Comments

Since (n+1)/2 is the mean of {1..n}, this sequence counts a type of "transitive" set partitions.

Links

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

Example

The a(1) = 1 through a(5) = 12 set partitions:
  {{1}}  {{12}}    {{123}}      {{1234}}        {{12345}}
         {{1}{2}}  {{13}{2}}    {{12}{34}}      {{1245}{3}}
                   {{1}{2}{3}}  {{13}{24}}      {{135}{24}}
                                {{14}{23}}      {{15}{234}}
                                {{1}{23}{4}}    {{1}{234}{5}}
                                {{14}{2}{3}}    {{12}{3}{45}}
                                {{1}{2}{3}{4}}  {{135}{2}{4}}
                                                {{14}{25}{3}}
                                                {{15}{24}{3}}
                                                {{1}{24}{3}{5}}
                                                {{15}{2}{3}{4}}
                                                {{1}{2}{3}{4}{5}}
The set partition {{1,3},{2,4}} has means {2,3}, with mean 5/2, so is counted under a(4).
The set partition {{1,3,5},{2,4}} has means {3,3}, with mean 3, so is counted under a(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]], Mean[Join@@#]==Mean[Mean/@#]&]], {n, 8}]

Crossrefs

For median instead of mean we have A361863.

A000110 counts set partitions.

A308037 counts set partitions with integer mean block-size.

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

A327481 counts subsets by mean, A013580 by median.

A361865 counts set partitions with integer mean of means.

A361911 counts set partitions with integer sum of means.

Cf. A007837, A035470, A038041, A067538, A275714, A275780, A326512, A326513, A361864, A361866.

Sequence in context: A275311 A056294 A084423 * A068134 A249051 A329413

Adjacent sequences: A361907 A361908 A361909 * A361911 A361912 A361913

Keyword

nonn,more

Author

Gus Wiseman, Apr 14 2023

Status

approved