A369910 Number of pairs (p,q) of partitions of n such that the set of parts in q is a proper subset of the set of parts in p.

0, 0, 0, 1, 3, 4, 15, 20, 52, 83, 163, 246, 501, 727, 1295, 1994, 3375, 4969, 8267, 12036, 19287, 28270, 43511, 62799, 96364, 137358, 204388, 291607, 427446, 601257, 874088, 1218524, 1743989, 2424096, 3422084, 4718626, 6622937, 9053800, 12559895, 17112883

Offset: 0

Author

Alois P. Heinz, Feb 05 2024

Keywords

• nonn

Examples

			a(5) = 4: (2111, 11111), (221, 11111), (311, 11111), (41, 11111).
a(6) = 15: (21111, 111111), (21111, 222), (2211, 111111), (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. A297388, A369704, A369707.

Programs

Maple

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

Mathematica

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