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

1,2

Comments

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

Links

Table of n, a(n) for n=1..70.

Example

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)

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]}]

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 partitions, 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.

Sequence in context: A154279 A065370 A147783 * A214340 A283463 A283464

Adjacent sequences: A371780 A371781 A371782 * A371784 A371785 A371786

Keyword

nonn,tabf

Author

Gus Wiseman, Apr 14 2024

Extensions

More terms from Jinyuan Wang, Feb 13 2025

Name edited by Peter Munn, Mar 05 2025

Status

approved