A371783 Irregular triangle read by rows where T(n,d) is the number of integer partitions of n that can be partitioned into d blocks with equal sums, with d ranging over all divisors d|n.

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

Offset

1,2

Comments

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

Links

Table of n, a(n) for n=1..45.

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 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, 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,more

Author

Gus Wiseman, Apr 14 2024

Status

approved