A369707 - OEIS

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

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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];