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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.

%I A369910 #13 Feb 29 2024 13:00:51
%S A369910 0,0,0,1,3,4,15,20,52,83,163,246,501,727,1295,1994,3375,4969,8267,
%T A369910 12036,19287,28270,43511,62799,96364,137358,204388,291607,427446,
%U A369910 601257,874088,1218524,1743989,2424096,3422084,4718626,6622937,9053800,12559895,17112883
%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.
%H A369910 Alois P. Heinz, <a href="/A369910/b369910.txt">Table of n, a(n) for n = 0..350</a>
%e A369910 a(5) = 4: (2111, 11111), (221, 11111), (311, 11111), (41, 11111).
%e A369910 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).
%p A369910 b:= proc(n, m, i, t) option remember; `if`(n=0,
%p A369910      `if`(t and m=0, 1, 0), `if`(i<1, 0, b(n, m, i-1, t)+add(
%p A369910       add(b(n-i*j, m-i*h, i-1, h=0 or t), h=0..m/i), j=1..n/i)))
%p A369910     end:
%p A369910 a:= n-> b(n$3, false):
%p A369910 seq(a(n), n=0..42);
%t A369910 b[n_, m_, i_, t_] := b[n, m, i, t] = If[n == 0,
%t A369910    If[t && m == 0, 1, 0], If[i < 1, 0, b[n, m, i-1, t] +
%t A369910    Sum[Sum[b[n-i*j, m-i*h, i-1, h == 0 || t], {h, 0, m/i}], {j, 1, n/i}]]];
%t A369910 a[n_] := b[n, n, n, False];
%t A369910 Table[a[n], {n, 0, 42}] (* _Jean-François Alcover_, Feb 29 2024, after _Alois P. Heinz_ *)
%Y A369910 Cf. A297388, A369704, A369707.
%K A369910 nonn
%O A369910 0,5
%A A369910 _Alois P. Heinz_, Feb 05 2024