A369707 Number of pairs (p,q) of distinct partitions of n such that the set of parts in q is a subset of the set of parts in p.

0, 0, 0, 1, 3, 6, 17, 28, 62, 107, 201, 316, 607, 909, 1567, 2444, 4025, 5979, 9749, 14250, 22467, 32950, 50137, 72295, 109728, 156182, 230570, 328089, 477606, 670213, 968324, 1346662, 1917385, 2658120, 3736326, 5139004, 7183707, 9798418, 13546453, 18414693

Offset: 0

Author

Alois P. Heinz, Jan 29 2024

Keywords

• nonn

Examples

			a(5) = 6: (2111, 11111), (2111, 221), (221, 11111), (221, 2111), (311, 11111), (41, 11111).
a(6) = 17: (21111, 111111), (21111, 2211), (21111, 222), (2211, 111111), (2211, 21111), (2211, 222), (3111, 111111), (321, 111111), (321, 21111), (321, 2211), (321, 222), (321, 3111), (3111, 33), (321, 33), (411, 111111), (42, 222), (51, 111111).
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..350

Crossrefs

Cf. A000041, A369704, A369910.

Programs

Maple

b:= proc(n, m, i) option remember; `if`(n=0,
     `if`(m=0, 1, 0), `if`(i<1, 0, b(n, m, i-1)+add(
      add(b(n-i*j, m-i*h, i-1), h=0..m/i), j=1..n/i)))
    end:
a:= n-> b(n$3)-combinat[numbpart](n):
seq(a(n), n=0..42);

Mathematica

b[n_, m_, i_] := b[n, m, i] = If[n == 0,
    If[m == 0, 1, 0], If[i < 1, 0, b[n, m, i - 1] +
    Sum[Sum[b[n - i*j, m - i*h, i - 1], {h, 0, m/i}], {j, 1, n/i}]]];
a[n_] := b[n, n, n] - PartitionsP[n];
Table[a[n], {n, 0, 42}] (* Jean-François Alcover, Feb 29 2024, after Alois P. Heinz *)

Formula

a(n) = A369704(n) - A000041(n).