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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.
3
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
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
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Jan 29 2024
STATUS
approved