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

0,5

Links

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

Example

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).

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 *)

Crossrefs

Cf. A297388, A369704, A369707.

Sequence in context: A041435 A136210 A041819 * A095799 A109926 A272514

Adjacent sequences: A369907 A369908 A369909 * A369911 A369912 A369913

Keyword

nonn

Author

Alois P. Heinz, Feb 05 2024

Status

approved