A361865 - OEIS

%I #6 Apr 05 2023 08:29:49

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

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

%e 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).

%e The a(1) = 1 through a(5) = 12 set partitions:

%e   {{1}}  .  {{123}}      {{1}{234}}  {{12345}}

%e             {{13}{2}}    {{123}{4}}  {{1245}{3}}

%e             {{1}{2}{3}}              {{135}{24}}

%e                                      {{15}{234}}

%e                                      {{1}{234}{5}}

%e                                      {{12}{3}{45}}

%e                                      {{135}{2}{4}}

%e                                      {{14}{25}{3}}

%e                                      {{15}{24}{3}}

%e                                      {{1}{24}{3}{5}}

%e                                      {{15}{2}{3}{4}}

%e                                      {{1}{2}{3}{4}{5}}

%t sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]& /@ sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];

%t Table[Length[Select[sps[Range[n]],IntegerQ[Mean[Mean/@#]]&]],{n,6}]

%Y For median instead of mean we have A361864.

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

%Y A000110 counts set partitions.

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

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

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

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

%K nonn,more

%O 1,3

%A _Gus Wiseman_, Apr 04 2023