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

0,5

Links

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

Formula

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

Example

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).

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 *)

Crossrefs

Cf. A000041, A369704, A369910.

Sequence in context: A024823 A024315 A235088 * A327068 A307604 A049943

Adjacent sequences: A369704 A369705 A369706 * A369708 A369709 A369710

Keyword

nonn

Author

Alois P. Heinz, Jan 29 2024

Status

approved