A369704 Number of pairs (p,q) of partitions of n such that the set of parts in q is a subset of the set of parts in p.

1, 1, 2, 4, 8, 13, 28, 43, 84, 137, 243, 372, 684, 1010, 1702, 2620, 4256, 6276, 10134, 14740, 23094, 33742, 51139, 73550, 111303, 158140, 233006, 331099, 481324, 674778, 973928, 1353504, 1925734, 2668263, 3748636, 5153887, 7201684, 9820055, 13572468, 18445878

Offset

0,3

Links

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

Formula

a(n) = A000041(n) + A369707(n).

Example

a(5) = 13: (11111, 11111), (2111, 11111), (2111, 2111), (2111, 221), (221, 11111), (221, 2111), (221, 221), (311, 11111), (311, 311), (32, 32), (41, 11111), (41, 41), (5, 5).

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):
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];
Table[a[n], {n, 0, 42}] (* Jean-François Alcover, Feb 29 2024, after Alois P. Heinz *)

Crossrefs

Cf. A000041, A001255, A077285, A297388, A369707, A369910.

Sequence in context: A288925 A018285 A026665 * A174540 A354687 A253766

Adjacent sequences: A369701 A369702 A369703 * A369705 A369706 A369707

Keyword

nonn

Author

Alois P. Heinz, Jan 29 2024

Status

approved