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A355389
Number of unordered pairs of distinct integer partitions of n.
4
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
0,4
FORMULA
a(n) = binomial(A000041(n), 2) = A355390(n)/2.
EXAMPLE
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)
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}]
PROG
(PARI) a(n) = binomial(numbpart(n), 2); \\ Michel Marcus, Jul 05 2022
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.
Sequence in context: A006308 A008837 A176098 * A081950 A204340 A331017
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jul 04 2022
STATUS
approved