A372122 Number of strict triquanimous partitions of 3n.

0, 0, 0, 0, 0, 1, 1, 4, 5, 13, 18, 36, 51, 93

Offset

0,8

Comments

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.

Links

Table of n, a(n) for n=0..13.

Example

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).
The a(5) = 1 through a(9) = 13 partitions:
  (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)
                            (7,6,4,3,1)    (8,7,5,3,1)    (9,7,5,4,2)
                            (7,6,5,2,1)    (8,7,6,2,1)    (9,7,6,3,2)
                            (6,5,4,3,2,1)  (7,6,5,3,2,1)  (9,8,5,4,1)
                                           (8,6,4,3,2,1)  (9,8,6,3,1)
                                                          (9,8,7,2,1)
                                                          (7,6,5,4,3,2)
                                                          (8,6,5,4,3,1)
                                                          (8,7,5,4,2,1)
                                                          (8,7,6,3,2,1)
                                                          (9,6,5,4,2,1)
                                                          (9,7,5,3,2,1)
                                                          (9,8,4,3,2,1)

Mathematica

hwt[n_]:=Total[Cases[FactorInteger[n], {p_, k_}:>PrimePi[p]*k]];
facs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[facs[n/d], Min@@#>=d&]], {d, Rest[Divisors[n]]}]];
Table[Length[Select[IntegerPartitions[3n], UnsameQ@@#&&Select[facs[Times@@Prime/@#], Length[#]==3&&SameQ@@hwt/@#&]!={}&]], {n, 0, 10}]

Crossrefs

The non-strict biquanimous version is A002219, ranks A357976.

The non-strict version is A002220, ranks A371955.

The biquanimous version is A237258, ranks A357854.

A321451 counts non-quanimous partitions, ranks A321453.

A321452 counts quanimous partitions, ranks A321454, strict A371737.

A371783 counts k-quanimous partitions.

A371795 counts non-biquanimous partitions, even case A006827, ranks A371731.

Cf. A035470, A064914, A321142, A321455, A371781, A371796.

Sequence in context: A191136 A083709 A091183 * A234254 A258281 A094029

Adjacent sequences: A372119 A372120 A372121 * A372123 A372124 A372125

Keyword

nonn,more

Author

Gus Wiseman, Apr 20 2024

Status

approved