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A265506 Number of pairs (p,q) of partitions of n into distinct parts such that p majorizes q in the dominance order.

%I A265506 #13 Feb 06 2017 12:11:25
%S A265506 1,1,1,3,3,6,10,15,21,35,54,75,115,161,238,349,486,673,972,1323,1840,
%T A265506 2562,3478,4711,6407,8624,11533,15502,20574,27194,36030,47320,61833,
%U A265506 81139,105286,136845,177369,228563,293787,377803,483090,616546,785925,997987
%N A265506 Number of pairs (p,q) of partitions of n into distinct parts such that p majorizes q in the dominance order.
%H A265506 Alois P. Heinz, <a href="/A265506/b265506.txt">Table of n, a(n) for n = 0..250</a>
%H A265506 Wikipedia, <a href="https://en.wikipedia.org/wiki/Dominance_order">Dominance Order</a>
%e A265506 a(3) = 3: (21,21), (3,21), (3,3).
%e A265506 a(4) = 3: (31,31), (4,31), (4,4).
%e A265506 a(5) = 6: (32,32), (41,32), (41,41), (5,32), (5,41), (5,5).
%e A265506 a(6) = 10: (321,321), (42,321), (42,42), (51,321), (51,42), (51,51), (6,321), (6,42), (6,51), (6,6).
%p A265506 b:= proc(n, m, i, j, t) option remember; `if`(n<m, 0, `if`(n=0, 1,
%p A265506       `if`(i<1, 0, `if`(t and j>0, b(n, m, i, j-1, true), 0)+
%p A265506       b(n, m, i-1, j, false)+b(n-i, m-j, max(0, min(n-i, i-1)),
%p A265506       max(0, min(m-j, j-1)), true))))
%p A265506     end:
%p A265506 a:= n-> b(n$4, true):
%p A265506 seq(a(n), n=0..40);
%t A265506 b[n_, m_, i_, j_, t_] := b[n, m, i, j, t] = If[n < m, 0, If[n == 0, 1, If[i < 1, 0, If[t && j > 0, b[n, m, i, j-1, True], 0] + b[n, m, i-1, j, False] + b[n-i, m-j, Max[0, Min[n-i, i-1]], Max[0, Min[m-j, j-1]], True]]]]; a[n_] := b[n, n, n, n, True]; Table[a[n], {n, 0, 40}] (* _Jean-François Alcover_, Feb 05 2017, translated from Maple *)
%Y A265506 Cf. A000009, A182988, A265508.
%K A265506 nonn
%O A265506 0,4
%A A265506 _Alois P. Heinz_, Dec 09 2015