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

%I #7 Apr 15 2023 23:31:09

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

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

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

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

%e {{1}} {{12}} {{123}} {{1234}} {{12345}}

%e {{1}{2}} {{13}{2}} {{12}{34}} {{1245}{3}}

%e {{1}{2}{3}} {{13}{24}} {{135}{24}}

%e {{14}{23}} {{15}{234}}

%e {{1}{23}{4}} {{1}{234}{5}}

%e {{14}{2}{3}} {{12}{3}{45}}

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

%e The set partition {{1,3},{2,4}} has means {2,3}, with mean 5/2, so is counted under a(4).

%e The set partition {{1,3,5},{2,4}} has means {3,3}, with mean 3, so is counted under a(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]],Mean[Join@@#]==Mean[Mean/@#]&]],{n,8}]

%Y For median instead of mean we have A361863.

%Y A000110 counts set partitions.

%Y A308037 counts set partitions with integer mean block-size.

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

%Y A327481 counts subsets by mean, A013580 by median.

%Y A361865 counts set partitions with integer mean of means.

%Y A361911 counts set partitions with integer sum of means.

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

%K nonn,more

%O 1,2

%A _Gus Wiseman_, Apr 14 2023