A361864 Number of set partitions of {1..n} whose block-medians have integer median.

1, 0, 3, 6, 30, 96, 461, 2000, 10727, 57092, 342348

Offset: 1

Author

Gus Wiseman, Apr 04 2023

Keywords

• nonn
• more

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) = 6 set partitions:
  {{1}}  .  {{123}}      {{1}{234}}
            {{13}{2}}    {{123}{4}}
            {{1}{2}{3}}  {{1}{2}{34}}
                         {{12}{3}{4}}
                         {{1}{24}{3}}
                         {{13}{2}{4}}
The set partition {{1,2},{3},{4}} has block-medians {3/2,3,4}, with median 3, so is counted under a(4).
		

Crossrefs

For mean instead of median we have A361865.

For sum instead of outer median we have A361911, means 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 counts partitions w/ integer median, complement A307683.

A360005 gives twice median of prime indices, distinct A360457.

Cf. A007837, A035470, A038041, A275714, A275780, A326512, A326513.

Cf. A027193, A067659, A079309, A231147, A359893, A359907, A361801.

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[Median[Median/@#]]&]],{n,6}]