cp's OEIS Frontend

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

%I A361864 #5 Apr 04 2023 07:41:38
%S A361864 1,0,3,6,30,96,461,2000,10727,57092,342348
%N A361864 Number of set partitions of {1..n} whose block-medians have integer median.
%C A361864 The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).
%e A361864 The a(1) = 1 through a(4) = 6 set partitions:
%e A361864   {{1}}  .  {{123}}      {{1}{234}}
%e A361864             {{13}{2}}    {{123}{4}}
%e A361864             {{1}{2}{3}}  {{1}{2}{34}}
%e A361864                          {{12}{3}{4}}
%e A361864                          {{1}{24}{3}}
%e A361864                          {{13}{2}{4}}
%e A361864 The set partition {{1,2},{3},{4}} has block-medians {3/2,3,4}, with median 3, so is counted under a(4).
%t A361864 sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]& /@ sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t A361864 Table[Length[Select[sps[Range[n]],IntegerQ[Median[Median/@#]]&]],{n,6}]
%Y A361864 For mean instead of median we have A361865.
%Y A361864 For sum instead of outer median we have A361911, means A361866.
%Y A361864 A000110 counts set partitions.
%Y A361864 A000975 counts subsets with integer median, mean A327475.
%Y A361864 A013580 appears to count subsets by median, A327481 by mean.
%Y A361864 A308037 counts set partitions with integer average block-size.
%Y A361864 A325347 counts partitions w/ integer median, complement A307683.
%Y A361864 A360005 gives twice median of prime indices, distinct A360457.
%Y A361864 Cf. A007837, A035470, A038041, A275714, A275780, A326512, A326513.
%Y A361864 Cf. A027193, A067659, A079309, A231147, A359893, A359907, A361801.
%K A361864 nonn,more
%O A361864 1,3
%A A361864 _Gus Wiseman_, Apr 04 2023