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

Author

Gus Wiseman, Apr 14 2023

Keywords

• nonn

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).

Examples

			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).
		

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.

Programs

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}]

Extensions

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