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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.

%I A369707 #17 Feb 29 2024 14:25:12
%S A369707 0,0,0,1,3,6,17,28,62,107,201,316,607,909,1567,2444,4025,5979,9749,
%T A369707 14250,22467,32950,50137,72295,109728,156182,230570,328089,477606,
%U A369707 670213,968324,1346662,1917385,2658120,3736326,5139004,7183707,9798418,13546453,18414693
%N 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.
%H A369707 Alois P. Heinz, <a href="/A369707/b369707.txt">Table of n, a(n) for n = 0..350</a>
%F A369707 a(n) = A369704(n) - A000041(n).
%e A369707 a(5) = 6: (2111, 11111), (2111, 221), (221, 11111), (221, 2111), (311, 11111), (41, 11111).
%e A369707 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).
%p A369707 b:= proc(n, m, i) option remember; `if`(n=0,
%p A369707      `if`(m=0, 1, 0), `if`(i<1, 0, b(n, m, i-1)+add(
%p A369707       add(b(n-i*j, m-i*h, i-1), h=0..m/i), j=1..n/i)))
%p A369707     end:
%p A369707 a:= n-> b(n$3)-combinat[numbpart](n):
%p A369707 seq(a(n), n=0..42);
%t A369707 b[n_, m_, i_] := b[n, m, i] = If[n == 0,
%t A369707     If[m == 0, 1, 0], If[i < 1, 0, b[n, m, i - 1] +
%t A369707     Sum[Sum[b[n - i*j, m - i*h, i - 1], {h, 0, m/i}], {j, 1, n/i}]]];
%t A369707 a[n_] := b[n, n, n] - PartitionsP[n];
%t A369707 Table[a[n], {n, 0, 42}] (* _Jean-François Alcover_, Feb 29 2024, after _Alois P. Heinz_ *)
%Y A369707 Cf. A000041, A369704, A369910.
%K A369707 nonn
%O A369707 0,5
%A A369707 _Alois P. Heinz_, Jan 29 2024