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Number of non-quanimous subsets of {1..n} containing n, meaning there is only one set partition with equal block-sums.
15

%I #6 Apr 18 2024 09:32:47

%S 1,2,3,6,11,21,40,77,144,279

%N Number of non-quanimous subsets of {1..n} containing n, meaning there is only one set partition with equal block-sums.

%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 not counted under a(9).

%e The a(1) = 1 through a(5) = 11 subsets:

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

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

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

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

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

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

%e {1,3,5}

%e {2,4,5}

%e {3,4,5}

%e {1,2,3,5}

%e {1,3,4,5}

%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]], MemberQ[#,n]&&Length[Select[sps[#],SameQ@@Total/@#&]]==1&]],{n,10}]

%Y First differences of A371789, complement counted by A371796.

%Y The "bi-" version is A371793, complement A232466.

%Y The complement is counted by A371797.

%Y A371736 counts non-quanimous strict partitions.

%Y A371737 counts quanimous strict partitions.

%Y A371783 counts k-quanimous partitions.

%Y A371791 counts biquanimous subsets, complement A371792.

%Y Cf. A002219, A035470, A038041, A275972, A316402, A321451, A321452.

%K nonn,more

%O 1,2

%A _Gus Wiseman_, Apr 17 2024