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

0,12

Links

Alois P. Heinz, Table of n, a(n) for n = 0..250

Wikipedia, Dominance Order

Formula

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

Example

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}.

Maple

b:= proc(n, m, i, j, t) option remember; `if`(n<m, 0, `if`(n=0, 1,
      `if`(i<1, 0, `if`(t and 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))))
    end:
g:= proc(n, i) option remember; `if`(i*(i+1)/2<n, 0,
      `if`(n=0, 1, g(n, i-1)+`if`(i>n, 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 *)

Crossrefs

Cf. A000009, A000217, A248476, A265506.

Sequence in context: A077285 A072523 A054473 * A327042 A006168 A250115

Adjacent sequences: A265505 A265506 A265507 * A265509 A265510 A265511

Keyword

nonn

Author

Alois P. Heinz, Dec 09 2015

Status

approved