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URL: https://oeis.org/A361910

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A361910
Number of set partitions of {1..n} such that the mean of the means of the blocks is (n+1)/2.
2
1, 2, 3, 7, 12, 47, 99, 430, 1379, 5613, 21416, 127303, 532201, 3133846, 18776715, 114275757, 737859014
OFFSET
1,2
COMMENTS
Since (n+1)/2 is the mean of {1..n}, this sequence counts a type of "transitive" set partitions.
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.
Sequence in context: A275311 A056294 A084423 * A068134 A249051 A329413
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Apr 14 2023
EXTENSIONS
a(13)-a(17) from Christian Sievers, May 12 2025
STATUS
approved