|
%I #7 Apr 21 2024 10:01:03
%S 0,0,0,0,0,1,1,4,5,13,18,36,51,93
%N Number of strict triquanimous partitions of 3n.
%C 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 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 The a(5) = 1 through a(9) = 13 partitions:
%e (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 (7,6,4,3,1) (8,7,5,3,1) (9,7,5,4,2)
%e (7,6,5,2,1) (8,7,6,2,1) (9,7,6,3,2)
%e (6,5,4,3,2,1) (7,6,5,3,2,1) (9,8,5,4,1)
%e (8,6,4,3,2,1) (9,8,6,3,1)
%e (9,8,7,2,1)
%e (7,6,5,4,3,2)
%e (8,6,5,4,3,1)
%e (8,7,5,4,2,1)
%e (8,7,6,3,2,1)
%e (9,6,5,4,2,1)
%e (9,7,5,3,2,1)
%e (9,8,4,3,2,1)
%t hwt[n_]:=Total[Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]*k]];
%t facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&, Select[facs[n/d],Min@@#>=d&]], {d,Rest[Divisors[n]]}]];
%t Table[Length[Select[IntegerPartitions[3n], UnsameQ@@#&&Select[facs[Times@@Prime/@#], Length[#]==3&&SameQ@@hwt/@#&]!={}&]],{n,0,10}]
%Y The non-strict biquanimous version is A002219, ranks A357976.
%Y The non-strict version is A002220, ranks A371955.
%Y The biquanimous version is A237258, ranks A357854.
%Y A321451 counts non-quanimous partitions, ranks A321453.
%Y A321452 counts quanimous partitions, ranks A321454, strict A371737.
%Y A371783 counts k-quanimous partitions.
%Y A371795 counts non-biquanimous partitions, even case A006827, ranks A371731.
%Y Cf. A035470, A064914, A321142, A321455, A371781, A371796.
%K nonn,more
%O 0,8
%A _Gus Wiseman_, Apr 20 2024
|
