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A371737 Number of quanimous strict integer partitions of n, meaning there is more than one set partition with all equal block-sums.

%I A371737 #7 Apr 16 2024 19:21:50
%S A371737 0,0,0,0,0,0,1,0,1,0,3,0,4,0,7,1,9,0,16,0,21,4,32,0,45,0,63,13,84,0,
%T A371737 126,0,158,36,220,0,303,0,393,93,511,0,708,0,881,229,1156,0,1539,0,
%U A371737 1925,516,2445,0,3233,6,3952,1134,5019,0,6497
%N A371737 Number of quanimous strict integer partitions of n, meaning there is more than one set partition with all equal block-sums.
%C A371737 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.
%C A371737 Conjecture: (1) Positions of 0's are A327782. (2) Positions of terms > 0 are A368459.
%e A371737 The a(0) = 0 through a(14) = 7 strict partitions:
%e A371737   .  .  .  .  .  .  (321)  .  (431)  .  (532)   .  (642)   .  (743)
%e A371737                                         (541)      (651)      (752)
%e A371737                                         (4321)     (5421)     (761)
%e A371737                                                    (6321)     (5432)
%e A371737                                                               (6431)
%e A371737                                                               (6521)
%e A371737                                                               (7421)
%t A371737 sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]& /@ sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t A371737 Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&Length[Select[sps[#], SameQ@@Total/@#&]]>1&]],{n,0,30}]
%Y A371737 The non-strict "bi-" version is A002219, ranks A357976.
%Y A371737 The "bi-" version is A237258, ranks A357854, complement A321142 or A371794.
%Y A371737 The non-strict version is A321452, ranks A321454.
%Y A371737 The complement is A371736, non-strict A321451, ranks A321453.
%Y A371737 The non-strict "bi-" complement is A371795, ranks A371731.
%Y A371737 A371783 counts k-quanimous partitions.
%Y A371737 A371791 counts biquanimous sets, complement A371792.
%Y A371737 A371796 counts quanimous sets, complement A371789.
%Y A371737 Cf. A000005, A018818, A035470, A038041, A064688, A232466, A237194, A305551, A365663, A365661, A365925, A371733, A371839.
%K A371737 nonn
%O A371737 0,11
%A A371737 _Gus Wiseman_, Apr 14 2024