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

%I A361911 #11 Aug 26 2024 20:08:14
%S A361911 1,1,3,10,30,107,479,2249,11173,60144,351086,2171087,14138253,
%T A361911 97097101,701820663,5303701310,41838047938,343716647215,2935346815495,
%U A361911 25999729551523,238473713427285,2261375071834708,22141326012712122,223519686318676559,2323959300370456901
%N A361911 Number of set partitions of {1..n} with block-medians summing to an integer.
%C A361911 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 A361911 The a(1) = 1 through a(4) = 10 set partitions:
%e A361911   {{1}}  {{1}{2}}  {{123}}      {{1}{234}}
%e A361911                    {{13}{2}}    {{12}{34}}
%e A361911                    {{1}{2}{3}}  {{123}{4}}
%e A361911                                 {{124}{3}}
%e A361911                                 {{13}{24}}
%e A361911                                 {{134}{2}}
%e A361911                                 {{14}{23}}
%e A361911                                 {{1}{24}{3}}
%e A361911                                 {{13}{2}{4}}
%e A361911                                 {{1}{2}{3}{4}}
%e A361911 The set partition {{1,4},{2,3}} has medians {5/2,5/2}, with sum 5, so is counted under a(4).
%t A361911 sps[{}]:={{}}; sps[set:{i_,___}] := Join@@Function[s,Prepend[#,s]& /@ sps[Complement[set,s]]] /@ Cases[Subsets[set],{i,___}];
%t A361911 Table[Length[Select[sps[Range[n]], IntegerQ[Total[Median/@#]]&]],{n,10}]
%Y A361911 For median instead of sum we have A361864.
%Y A361911 For mean of means we have A361865.
%Y A361911 For mean instead of median we have A361866.
%Y A361911 A000110 counts set partitions.
%Y A361911 A000975 counts subsets with integer median, mean A327475.
%Y A361911 A013580 appears to count subsets by median, A327481 by mean.
%Y A361911 A308037 counts set partitions with integer average block-size.
%Y A361911 A325347 = partitions w/ integer median, complement A307683, strict A359907.
%Y A361911 A360005 gives twice median of prime indices, distinct A360457.
%Y A361911 Cf. A007837, A035470, A038041, A079309, A231147, A275714, A275780, A359893, A361801, A361910.
%K A361911 nonn
%O A361911 1,3
%A A361911 _Gus Wiseman_, Apr 14 2023
%E A361911 a(12)-a(25) from _Christian Sievers_, Aug 26 2024