cp's OEIS Frontend

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

%I A361910 #10 May 12 2025 20:18:29
%S A361910 1,2,3,7,12,47,99,430,1379,5613,21416,127303,532201,3133846,18776715,
%T A361910 114275757,737859014
%N A361910 Number of set partitions of {1..n} such that the mean of the means of the blocks is (n+1)/2.
%C A361910 Since (n+1)/2 is the mean of {1..n}, this sequence counts a type of "transitive" set partitions.
%e A361910 The a(1) = 1 through a(5) = 12 set partitions:
%e A361910   {{1}}  {{12}}    {{123}}      {{1234}}        {{12345}}
%e A361910          {{1}{2}}  {{13}{2}}    {{12}{34}}      {{1245}{3}}
%e A361910                    {{1}{2}{3}}  {{13}{24}}      {{135}{24}}
%e A361910                                 {{14}{23}}      {{15}{234}}
%e A361910                                 {{1}{23}{4}}    {{1}{234}{5}}
%e A361910                                 {{14}{2}{3}}    {{12}{3}{45}}
%e A361910                                 {{1}{2}{3}{4}}  {{135}{2}{4}}
%e A361910                                                 {{14}{25}{3}}
%e A361910                                                 {{15}{24}{3}}
%e A361910                                                 {{1}{24}{3}{5}}
%e A361910                                                 {{15}{2}{3}{4}}
%e A361910                                                 {{1}{2}{3}{4}{5}}
%e A361910 The set partition {{1,3},{2,4}} has means {2,3}, with mean 5/2, so is counted under a(4).
%e A361910 The set partition {{1,3,5},{2,4}} has means {3,3}, with mean 3, so is counted under a(5).
%t A361910 sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]& /@ sps[Complement[set,s]]] /@ Cases[Subsets[set],{i,___}];
%t A361910 Table[Length[Select[sps[Range[n]],Mean[Join@@#]==Mean[Mean/@#]&]],{n,8}]
%Y A361910 For median instead of mean we have A361863.
%Y A361910 A000110 counts set partitions.
%Y A361910 A308037 counts set partitions with integer mean block-size.
%Y A361910 A327475 counts subsets with integer mean, A000975 with integer median.
%Y A361910 A327481 counts subsets by mean, A013580 by median.
%Y A361910 A361865 counts set partitions with integer mean of means.
%Y A361910 A361911 counts set partitions with integer sum of means.
%Y A361910 Cf. A007837, A035470, A038041, A067538, A275714, A275780, A326512, A326513, A361864, A361866.
%K A361910 nonn,more
%O A361910 1,2
%A A361910 _Gus Wiseman_, Apr 14 2023
%E A361910 a(13)-a(17) from _Christian Sievers_, May 12 2025