A365662 Number of ordered pairs of disjoint strict integer partitions of n.

1, 0, 0, 2, 2, 6, 8, 14, 18, 32, 42, 66, 92, 136, 190, 280, 374, 532, 744, 1014, 1366, 1896, 2512, 3384, 4526, 6006, 7910, 10496, 13648, 17842, 23338, 30116, 38826, 50256, 64298, 82258, 105156, 133480, 169392, 214778, 270620, 340554, 428772, 536302, 670522

Offset: 0

Author

Gus Wiseman, Sep 19 2023

Keywords

• nonn

Comments

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

Examples

			The a(0) = 1 through a(7) = 14 pairs:
  ()()  .  .  (21)(3)  (31)(4)  (32)(5)   (42)(6)   (43)(7)
              (3)(21)  (4)(31)  (41)(5)   (51)(6)   (52)(7)
                                (5)(32)   (6)(42)   (61)(7)
                                (5)(41)   (6)(51)   (7)(43)
                                (32)(41)  (321)(6)  (7)(52)
                                (41)(32)  (42)(51)  (7)(61)
                                          (51)(42)  (421)(7)
                                          (6)(321)  (43)(52)
                                                    (43)(61)
                                                    (52)(43)
                                                    (52)(61)
                                                    (61)(43)
                                                    (61)(52)
                                                    (7)(421)
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Crossrefs

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

If the partitions can have different sums we get A032302.

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

The unordered version is A108796, non-strict A260669.

A000041 counts integer partitions, strict A000009.

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

A000712 counts distinct submultisets of partitions.

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

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

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

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

Programs

Mathematica

Table[Length[Select[Tuples[Select[IntegerPartitions[n], UnsameQ@@#&],2], Intersection@@#=={}&]], {n,0,15}]
Table[SeriesCoefficient[Product[(1 + x^k + y^k), {k, 1, n}], {x, 0, n}, {y, 0, n}], {n, 0, 50}] (* Vaclav Kotesovec, Apr 24 2025 *)

Formula

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

a(n) = [(x*y)^n] Product_{k>=1} (1 + x^k + y^k). - Ilya Gutkovskiy, Apr 24 2025