A355389 Number of unordered pairs of distinct integer partitions of n.

0, 0, 1, 3, 10, 21, 55, 105, 231, 435, 861, 1540, 2926, 5050, 9045, 15400, 26565, 43956, 73920, 119805, 196251, 313236, 501501, 786885, 1239525, 1915903, 2965830, 4528545, 6909903, 10417330, 15699606, 23403061, 34848726, 51435153, 75761895, 110744403, 161577276

Offset: 0

Author

Gus Wiseman, Jul 04 2022

Keywords

• nonn

Examples

			The a(0) = 0 through a(4) = 10 pairs:
  .  .  (2)(11)  (3)(21)    (4)(22)
                 (3)(111)   (4)(31)
                 (21)(111)  (22)(31)
                            (4)(211)
                            (22)(211)
                            (31)(211)
                            (4)(1111)
                            (22)(1111)
                            (31)(1111)
                            (211)(1111)
		

Crossrefs

The version for compositions is A006516.

Without distinctness we get A086737.

The unordered version is A355390, without distinctness A001255.

A000041 counts partitions, strict A000009.

A001970 counts multiset partitions of partitions.

A063834 counts partitions of each part of a partition.

Cf. A022811, A070933, A316245, A317715, A319794, A319910, A323433, A339006, A355385.

Programs

Maple

a:= n-> binomial(combinat[numbpart](n),2):
seq(a(n), n=0..36);  # Alois P. Heinz, Feb 07 2024

Mathematica

Table[Binomial[PartitionsP[n],2],{n,0,6}]

PARI

a(n) = binomial(numbpart(n), 2); \\ Michel Marcus, Jul 05 2022

Formula

a(n) = binomial(A000041(n), 2) = A355390(n)/2.