A361864 - OEIS

%I #5 Apr 04 2023 07:41:38

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

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

%C 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 The a(1) = 1 through a(4) = 6 set partitions:

%e   {{1}}  .  {{123}}      {{1}{234}}

%e             {{13}{2}}    {{123}{4}}

%e             {{1}{2}{3}}  {{1}{2}{34}}

%e                          {{12}{3}{4}}

%e                          {{1}{24}{3}}

%e                          {{13}{2}{4}}

%e The set partition {{1,2},{3},{4}} has block-medians {3/2,3,4}, with median 3, so is counted under a(4).

%t sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]& /@ sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];

%t Table[Length[Select[sps[Range[n]],IntegerQ[Median[Median/@#]]&]],{n,6}]

%Y For mean instead of median we have A361865.

%Y For sum instead of outer median we have A361911, means A361866.

%Y A000110 counts set partitions.

%Y A000975 counts subsets with integer median, mean A327475.

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

%Y A308037 counts set partitions with integer average block-size.

%Y A325347 counts partitions w/ integer median, complement A307683.

%Y A360005 gives twice median of prime indices, distinct A360457.

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

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

%K nonn,more

%O 1,3

%A _Gus Wiseman_, Apr 04 2023