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A372122 Number of strict triquanimous partitions of 3n.

%I A372122 #11 Mar 30 2025 04:28:04
%S A372122 0,0,0,0,0,1,1,4,5,13,18,36,51,93,132,229,315,516,735,1134,1575,2407,
%T A372122 3309,4878,6710,9690,13168,18744,25114,35050,47210,64503,85573,116445,
%U A372122 153328,205367,269383,356668,464268,610644,788274,1026330,1321017,1704309,2176054
%N A372122 Number of strict triquanimous partitions of 3n.
%C A372122 A finite multiset of numbers is defined to be triquanimous iff it can be partitioned into three multisets with equal sums. Triquanimous partitions are counted by A002220 and ranked by A371955.
%e A372122 The partition (11,7,5,4,3,2,1) has qualifying set partitions {{11},{4,7},{1,2,3,5}} and {{11},{1,3,7},{2,4,5}} so is counted under a(11).
%e A372122 The a(5) = 1 through a(9) = 13 partitions:
%e A372122   (5,4,3,2,1)  (6,5,4,2,1)  (7,5,4,3,2)    (8,6,5,3,2)    (9,6,5,4,3)
%e A372122                             (7,6,4,3,1)    (8,7,5,3,1)    (9,7,5,4,2)
%e A372122                             (7,6,5,2,1)    (8,7,6,2,1)    (9,7,6,3,2)
%e A372122                             (6,5,4,3,2,1)  (7,6,5,3,2,1)  (9,8,5,4,1)
%e A372122                                            (8,6,4,3,2,1)  (9,8,6,3,1)
%e A372122                                                           (9,8,7,2,1)
%e A372122                                                           (7,6,5,4,3,2)
%e A372122                                                           (8,6,5,4,3,1)
%e A372122                                                           (8,7,5,4,2,1)
%e A372122                                                           (8,7,6,3,2,1)
%e A372122                                                           (9,6,5,4,2,1)
%e A372122                                                           (9,7,5,3,2,1)
%e A372122                                                           (9,8,4,3,2,1)
%t A372122 hwt[n_]:=Total[Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]*k]];
%t A372122 facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&, Select[facs[n/d],Min@@#>=d&]], {d,Rest[Divisors[n]]}]];
%t A372122 Table[Length[Select[IntegerPartitions[3n], UnsameQ@@#&&Select[facs[Times@@Prime/@#], Length[#]==3&&SameQ@@hwt/@#&]!={}&]],{n,0,10}]
%Y A372122 The non-strict biquanimous version is A002219, ranks A357976.
%Y A372122 The non-strict version is A002220, ranks A371955.
%Y A372122 The biquanimous version is A237258, ranks A357854.
%Y A372122 A321451 counts non-quanimous partitions, ranks A321453.
%Y A372122 A321452 counts quanimous partitions, ranks A321454, strict A371737.
%Y A372122 A371783 counts k-quanimous partitions.
%Y A372122 A371795 counts non-biquanimous partitions, even case A006827, ranks A371731.
%Y A372122 Cf. A035470, A064914, A321142, A321455, A371781, A371796.
%K A372122 nonn
%O A372122 0,8
%A A372122 _Gus Wiseman_, Apr 20 2024
%E A372122 More terms from _Jinyuan Wang_, Mar 30 2025