A355385 Number of pairs (y, v) of integer partitions of n where the length of v equals the number of distinct parts in y.

1, 1, 2, 3, 7, 12, 25, 43, 81, 141, 243, 409, 699, 1132, 1844, 2995, 4744, 7408, 11655, 17839, 27509, 41546, 62879, 93537, 139974, 205547, 302714, 440097, 640968, 921774, 1327538, 1891548, 2696635, 3809860, 5380257, 7540778, 10561566, 14687109, 20408170, 28183998, 38882009

Offset: 0

Author

Gus Wiseman, Jul 02 2022

Keywords

• nonn

Comments

Also the number of composable pairs of integer partitions of n, where a partition is regarded as an arrow from (number of parts) to (number of distinct parts). Is there a nice choice of a composition operation making this into an associative category?

Examples

			The a(0) = 1 through a(5) = 10 pairs:
  ()()  (1)(1)  (2)(2)   (3)(3)    (4)(4)     (5)(5)
                (11)(2)  (21)(21)  (31)(31)   (41)(41)
                         (111)(3)  (31)(22)   (41)(32)
                                   (22)(4)    (32)(41)
                                   (211)(31)  (32)(32)
                                   (211)(22)  (311)(41)
                                   (1111)(4)  (311)(32)
                                              (221)(41)
                                              (221)(32)
                                              (2111)(41)
                                              (2111)(32)
                                              (11111)(5)
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..1000

Crossrefs

The inhomogeneous version with containment and multiplicity is A339006.

The inhomogeneous version with containment is A355383.

The inhomogeneous version with containment for compositions is A355384.

The version for compositions is A355388.

A001970 counts multiset partitions of partitions.

A063834 counts partitions of each part of a partition.

A323583 counts splittings of partitions.

Cf. A000009, A008284, A022811, A032020, A070933, A072706, A116608, A279787, A319910, A336135.

Programs

Mathematica

Table[Length[Select[Tuples[IntegerPartitions[n],2],Length[Union[#[[1]]]]==Length[#[[2]]]&]],{n,0,15}]

PARI

\\ P gives A008284 and R gives A116608 as g.f.'s.
P(n,y) = {1/prod(k=1, n, 1 - y*x^k + O(x*x^n))}
R(n,y) = {prod(k=1, n, 1 + y/(1 - x^k) - y + O(x*x^n))}
seq(n) = {my(g=Vec(P(n,y)), h=Vec(R(n,y))); vector(n+1, i, my(p=g[i], q=h[i]); sum(j=0, poldegree(q), polcoef(p,j)*polcoef(q,j)))} \\ Andrew Howroyd, Dec 31 2022

Formula

a(n) = Sum_{j >= 1} A116608(n,j) * A008284(n,j) for n > 0. - Andrew Howroyd, Dec 31 2022

Extensions

Terms a(26) and beyond from Andrew Howroyd, Dec 31 2022