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, 532201, 3133846, 18776715, 114275757, 737859014

Offset: 1

Author

Gus Wiseman, Apr 14 2023

Keywords

• nonn
• more

Comments

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

Examples

			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).
		

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.

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]],Mean[Join@@#]==Mean[Mean/@#]&]],{n,8}]

Extensions

a(13)-a(17) from Christian Sievers, May 12 2025