A365662 - OEIS

A365662

Number of ordered pairs of disjoint strict integer partitions of n.

%I #6 Sep 20 2023 23:58:33

%S 1,0,0,2,2,6,8,14,18,32,42,66,92,136,190,280,374,532,744,1014,1366,

%T 1896,2512,3384,4526,6006,7910,10496,13648,17842,23338,30116,38826,

%U 50256,64298,82258,105156,133480,169392,214778,270620,340554,428772,536302,670522

%N Number of ordered pairs of disjoint strict integer partitions of n.

%C Also the number of ways to first choose a strict partition of 2n, then a subset of it summing to n.

%F a(n) = 2*A108796(n) for n > 1.

%e The a(0) = 1 through a(7) = 14 pairs:

%e ()() . . (21)(3) (31)(4) (32)(5) (42)(6) (43)(7)

%e (3)(21) (4)(31) (41)(5) (51)(6) (52)(7)

%e (5)(32) (6)(42) (61)(7)

%e (5)(41) (6)(51) (7)(43)

%e (32)(41) (321)(6) (7)(52)

%e (41)(32) (42)(51) (7)(61)

%e (51)(42) (421)(7)

%e (6)(321) (43)(52)

%e (43)(61)

%e (52)(43)

%e (52)(61)

%e (61)(43)

%e (61)(52)

%e (7)(421)

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

%Y For subsets instead of partitions we have A000244, non-disjoint A000302.

%Y If the partitions can have different sums we get A032302.

%Y The non-strict version is A054440, non-disjoint A001255.

%Y The unordered version is A108796, non-strict A260669.

%Y A000041 counts integer partitions, strict A000009.

%Y A000124 counts distinct possible sums of subsets of {1..n}.

%Y A000712 counts distinct submultisets of partitions.

%Y A002219 and A237258 count partitions of 2n including a partition of n.

%Y A304792 counts subset-sums of partitions, positive A276024, strict A284640.

%Y A364272 counts sum-full strict partitions, sum-free A364349.

%Y Cf. A006827, A046663, A064914, A122768, A260664, A365661, A365663.

%K nonn

%O 0,4

%A _Gus Wiseman_, Sep 19 2023