A265508 Number of unordered pairs {p,q} of partitions of n into distinct parts such that p and q are incomparable in the dominance order.

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 3, 5, 10, 15, 29, 42, 68, 109, 162, 240, 364, 527, 749, 1096, 1529, 2162, 3026, 4179, 5702, 7926, 10650, 14412, 19437, 26042, 34560, 46077, 60617, 79893, 104850, 136851, 177884, 231526, 298868, 385221, 496159, 635725, 812342

Offset: 0

Author

Alois P. Heinz, Dec 09 2015

Keywords

• nonn

Examples

			a(9) = 1: {621,54}.
a(10) = 1: {721,64}.
a(11) = 3: {821,74}, {821,65}, {731,65}.
a(12) = 5: {6321,543}, {921,84}, {921,75}, {831,75}, {732,651}.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..250
• Wikipedia, Dominance Order

Crossrefs

Cf. A000009, A000217, A248476, A265506.

Programs

Maple

b:= proc(n, m, i, j, t) option remember; `if`(n0, 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))))
    end:
g:= proc(n, i) option remember; `if`(i*(i+1)/2n, 0, g(n-i, i-1))))
    end:
a:= n-> (t-> t*(t+1)/2)(g(n$2))-b(n$4, true):
seq(a(n), n=0..45);

Mathematica

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 *)

Formula

a(n) = A000217(A000009(n)) - A265506(n).