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Number of ordered pairs of distinct integer partitions of n.
1

%I #10 Jul 05 2022 06:47:25

%S 0,0,2,6,20,42,110,210,462,870,1722,3080,5852,10100,18090,30800,53130,

%T 87912,147840,239610,392502,626472,1003002,1573770,2479050,3831806,

%U 5931660,9057090,13819806,20834660,31399212,46806122,69697452,102870306,151523790,221488806

%N Number of ordered pairs of distinct integer partitions of n.

%F a(n) = 2*A355389(n) = 2*binomial(A000041(n), 2).

%e The a(0) = 0 through a(3) = 6 pairs:

%e . . (11)(2) (21)(3)

%e (2)(11) (3)(21)

%e (111)(3)

%e (3)(111)

%e (111)(21)

%e (21)(111)

%t Table[Length[Select[Tuples[IntegerPartitions[n],2],UnsameQ@@#&]],{n,0,15}]

%o (PARI) a(n) = 2*binomial(numbpart(n), 2); \\ _Michel Marcus_, Jul 05 2022

%Y Without distinctness we have A001255, unordered A086737.

%Y The version for compositions is A020522, unordered A006516.

%Y The unordered version is A355389.

%Y A000041 counts partitions, strict A000009.

%Y A001970 counts multiset partitions of partitions.

%Y A063834 counts partitions of each part of a partition.

%Y Cf. A022811, A070933, A316245, A317715, A319910, A323433, A323583, A339006, A355385.

%K nonn

%O 0,3

%A _Gus Wiseman_, Jul 04 2022