A369707 -id:A369707 - cp's OEIS frontend

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

Alois P. Heinz, Jan 29 2024

nonn

			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).

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

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

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);

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 *)

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

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.

Original entry on oeis.org

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

Alois P. Heinz, Feb 05 2024

nonn

			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).

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

Cf. A297388, A369704, A369707.

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);

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 *)

cp's OEIS Frontend

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.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula