A371783 Irregular triangle read by rows where T(n,k) is the number of integer partitions of n that can be partitioned into d = A027750(n,k) blocks with equal sums.

1, 2, 1, 3, 1, 5, 3, 1, 7, 1, 11, 6, 4, 1, 15, 1, 22, 14, 5, 1, 30, 10, 1, 42, 25, 6, 1, 56, 1, 77, 53, 30, 15, 7, 1, 101, 1, 135, 89, 8, 1, 176, 65, 21, 1, 231, 167, 55, 9, 1, 297, 1, 385, 278, 173, 28, 10, 1, 490, 1, 627, 480, 140, 91, 11, 1, 792, 343, 36, 1

Offset: 1

Author

Gus Wiseman, Apr 14 2024

Keywords

• nonn
• tabf

Comments

These could be called d-quanimous partitions, cf. A002219, A064914, A321452.

Examples

			Triangle begins:
    1
    2   1
    3   1
    5   3   1
    7   1
   11   6   4   1
   15   1
   22  14   5   1
   30  10   1
   42  25   6   1
   56   1
   77  53  30  15   7   1
  101   1
  135  89   8   1
  176  65  21   1
Row n = 6 counts the following partitions:
  (6)       (33)      (222)     (111111)
  (33)      (321)     (2211)
  (42)      (2211)    (21111)
  (51)      (3111)    (111111)
  (222)     (21111)
  (321)     (111111)
  (411)
  (2211)
  (3111)
  (21111)
  (111111)
		

Crossrefs

Row lengths are A000005.

Column k = 1 is A000041.

Inserting zeros gives A371954.

Row sums are A372121.

A002219 (aerated) counts biquanimous partitions, ranks A357976.

A237258 aerated counts biquanimous strict partitions, ranks A357854.

A321142 and A371794 count non-biquanimous strict partitions.

A321451 counts non-quanimous partitions, ranks A321453.

A321452 counts quanimous partitions, ranks A321454.

A371736 counts non-quanimous strict partitons, complement A371737.

A371781 lists numbers with biquanimous prime signature, complement A371782.

A371789 counts non-quanimous sets, differences A371790.

A371796 counts quanimous sets, differences A371797.

Cf. A006827, A027750, A035470, A064914, A321455, A365543, A371791, 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,1,8},{k,Divisors[n]}]

Extensions

More terms from Jinyuan Wang, Feb 13 2025

Name edited by Peter Munn, Mar 05 2025