%I #7 Apr 17 2024 20:11:40

%S 0,0,0,1,3,8,19,43,94,206,439

%N Number of quanimous subsets of {1..n}, meaning there is more than one set partition with all equal block-sums.

%C A finite multiset of numbers is defined to be quanimous iff it can be partitioned into two or more multisets with equal sums. Quanimous partitions are counted by A321452 and ranked by A321454.

%e The set s = {3,4,6,8,9} has set partitions {{3,4,6,8,9}} and {{3,4,8},{6,9}} with equal block-sums, so s is counted under a(9).

%e The a(3) = 1 through a(6) = 19 subsets:

%e {1,2,3} {1,2,3} {1,2,3} {1,2,3}

%e {1,3,4} {1,3,4} {1,3,4}

%e {1,2,3,4} {1,4,5} {1,4,5}

%e {2,3,5} {1,5,6}

%e {1,2,3,4} {2,3,5}

%e {1,2,4,5} {2,4,6}

%e {2,3,4,5} {1,2,3,4}

%e {1,2,3,4,5} {1,2,3,6}

%e {1,2,4,5}

%e {1,2,5,6}

%e {1,3,4,6}

%e {2,3,4,5}

%e {2,3,5,6}

%e {3,4,5,6}

%e {1,2,3,4,5}

%e {1,2,3,4,6}

%e {1,2,4,5,6}

%e {2,3,4,5,6}

%e {1,2,3,4,5,6}

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

%t Table[Length[Select[Subsets[Range[n]], Length[Select[sps[#],SameQ@@Total/@#&]]>1&]],{n,0,10}]

%Y The "bi-" version for integer partitions is A002219 aerated, ranks A357976.

%Y The "bi-" version for strict partitions is A237258 aerated, ranks A357854.

%Y The complement for integer partitions is A321451, ranks A321453.

%Y The version for integer partitions is A321452, ranks A321454

%Y The version for strict partitions is A371737, complement A371736.

%Y The complement is counted by A371789, differences A371790.

%Y The "bi-" version is A371791, complement A371792.

%Y First differences are A371797.

%Y A108917 counts knapsack partitions, ranks A299702, strict A275972.

%Y A366754 counts non-knapsack partitions, ranks A299729, strict A316402.

%Y A371783 counts k-quanimous partitions.

%Y Cf. A000005, A018818, A035470, A038041, A232466, A279791, A321142, A365661, A371731, A371794, A371795.

%K nonn,more

%O 0,5

%A _Gus Wiseman_, Apr 17 2024