A361911 Number of set partitions of {1..n} with block-medians summing to an integer.

1, 1, 3, 10, 30, 107, 479, 2249, 11173, 60144, 351086, 2171087, 14138253, 97097101, 701820663, 5303701310, 41838047938, 343716647215, 2935346815495, 25999729551523, 238473713427285, 2261375071834708, 22141326012712122, 223519686318676559, 2323959300370456901

Offset

1,3

Comments

The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).

Links

Table of n, a(n) for n=1..25.

Example

The a(1) = 1 through a(4) = 10 set partitions:
  {{1}}  {{1}{2}}  {{123}}      {{1}{234}}
                   {{13}{2}}    {{12}{34}}
                   {{1}{2}{3}}  {{123}{4}}
                                {{124}{3}}
                                {{13}{24}}
                                {{134}{2}}
                                {{14}{23}}
                                {{1}{24}{3}}
                                {{13}{2}{4}}
                                {{1}{2}{3}{4}}
The set partition {{1,4},{2,3}} has medians {5/2,5/2}, with sum 5, so is counted under a(4).

Mathematica

sps[{}]:={{}}; sps[set:{i_, ___}] := Join@@Function[s, Prepend[#, s]& /@ sps[Complement[set, s]]] /@ Cases[Subsets[set], {i, ___}];
Table[Length[Select[sps[Range[n]], IntegerQ[Total[Median/@#]]&]], {n, 10}]

Crossrefs

For median instead of sum we have A361864.

For mean of means we have A361865.

For mean instead of median we have A361866.

A000110 counts set partitions.

A000975 counts subsets with integer median, mean A327475.

A013580 appears to count subsets by median, A327481 by mean.

A308037 counts set partitions with integer average block-size.

A325347 = partitions w/ integer median, complement A307683, strict A359907.

A360005 gives twice median of prime indices, distinct A360457.

Cf. A007837, A035470, A038041, A079309, A231147, A275714, A275780, A359893, A361801, A361910.

Sequence in context: A136846 A004663 A136853 * A215287 A316764 A331780

Adjacent sequences: A361908 A361909 A361910 * A361912 A361913 A361914

Keyword

nonn

Author

Gus Wiseman, Apr 14 2023

Extensions

a(12)-a(25) from Christian Sievers, Aug 26 2024

Status

approved