A371954 Triangle read by rows where T(n,k) is the number of integer partitions of n that can be partitioned into k multisets with equal sums (k-quanimous).

1, 0, 1, 0, 2, 1, 0, 3, 0, 1, 0, 5, 3, 0, 1, 0, 7, 0, 0, 0, 1, 0, 11, 6, 4, 0, 0, 1, 0, 15, 0, 0, 0, 0, 0, 1, 0, 22, 14, 0, 5, 0, 0, 0, 1, 0, 30, 0, 10, 0, 0, 0, 0, 0, 1, 0, 42, 25, 0, 0, 6, 0, 0, 0, 0, 1, 0, 56, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 77, 53, 30, 15, 0, 7, 0, 0, 0, 0, 0, 1

Offset: 0

Author

Gus Wiseman, Apr 20 2024

Keywords

• nonn
• tabl

Comments

A finite multiset of numbers is defined to be k-quanimous iff it can be partitioned into k multisets with equal sums.

Examples

			Triangle begins:
  1
  0  1
  0  2  1
  0  3  0  1
  0  5  3  0  1
  0  7  0  0  0  1
  0 11  6  4  0  0  1
  0 15  0  0  0  0  0  1
  0 22 14  0  5  0  0  0  1
  0 30  0 10  0  0  0  0  0  1
  0 42 25  0  0  6  0  0  0  0  1
  0 56  0  0  0  0  0  0  0  0  0  1
  0 77 53 30 15  0  7  0  0  0  0  0  1
Row n = 6 counts the following partitions:
  .  (6)       (33)      (222)     .  .  (111111)
     (51)      (321)     (2211)
     (42)      (3111)    (21111)
     (411)     (2211)    (111111)
     (33)      (21111)
     (321)     (111111)
     (3111)
     (222)
     (2211)
     (21111)
     (111111)
		

Crossrefs

Row n has A000005(n) positive entries.

Column k = 1 is A000041.

Column k = 2 is A002219 (aerated), ranks A357976.

Column k = 3 is A002220 (aerated), ranks A371955.

Removing all zeros gives A371783.

Row sums are A372121.

A321451 counts non-quanimous partitions, ranks A321453.

A321452 counts quanimous partitions, ranks A321454.

A371789 counts non-quanimous sets, complement A371796.

Cf. A002221, A002222, A006827, A035470, A064914, A237258, A321142, A321455, A365543, A371795.

Programs

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[n], Select[facs[Times@@Prime/@#], Length[#]==k&&SameQ@@hwt/@#&]!={}&]],{n,0,10},{k,0,n}]