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%I #13 Feb 06 2017 12:12:04
%S 0,0,0,0,0,0,0,0,0,1,1,3,5,10,15,29,42,68,109,162,240,364,527,749,
%T 1096,1529,2162,3026,4179,5702,7926,10650,14412,19437,26042,34560,
%U 46077,60617,79893,104850,136851,177884,231526,298868,385221,496159,635725,812342
%N Number of unordered pairs {p,q} of partitions of n into distinct parts such that p and q are incomparable in the dominance order.
%H Alois P. Heinz, <a href="/A265508/b265508.txt">Table of n, a(n) for n = 0..250</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Dominance_order">Dominance Order</a>
%F a(n) = A000217(A000009(n)) - A265506(n).
%e a(9) = 1: {621,54}.
%e a(10) = 1: {721,64}.
%e a(11) = 3: {821,74}, {821,65}, {731,65}.
%e a(12) = 5: {6321,543}, {921,84}, {921,75}, {831,75}, {732,651}.
%p b:= proc(n, m, i, j, t) option remember; `if`(n<m, 0, `if`(n=0, 1,
%p `if`(i<1, 0, `if`(t and j>0, b(n, m, i, j-1, true), 0)+
%p b(n, m, i-1, j, false)+b(n-i, m-j, max(0, min(n-i, i-1)),
%p max(0, min(m-j, j-1)), true))))
%p end:
%p g:= proc(n, i) option remember; `if`(i*(i+1)/2<n, 0,
%p `if`(n=0, 1, g(n, i-1)+`if`(i>n, 0, g(n-i, i-1))))
%p end:
%p a:= n-> (t-> t*(t+1)/2)(g(n$2))-b(n$4, true):
%p seq(a(n), n=0..45);
%t 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]]]]; g[n_, i_] := g[n, i] = If[i*(i+1)/2 < n, 0, If[n == 0, 1, g[n, i-1] + If[i > n, 0, g[n-i, i-1]]]]; a[n_] := (#*(#+1)/2&)[g[n, n]] - b[n, n, n, n, True]; Table[a[n], {n, 0, 45}] (* _Jean-François Alcover_, Feb 05 2017, translated from Maple *)
%Y Cf. A000009, A000217, A248476, A265506.
%K nonn
%O 0,12
%A _Alois P. Heinz_, Dec 09 2015
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