A248476 -id:A248476 - cp's OEIS frontend

A182988 The number of dominance pairs of integer partitions of n according to either/or dominance order, where dominance between two partitions x and y means that x is majorized by y or y is majorized by x.

1, 1, 4, 9, 25, 49, 117, 217, 454, 830, 1594, 2796, 5159, 8777, 15415, 25810, 43819, 71595, 118629, 190148, 307519, 485660, 769382, 1195807, 1864617, 2857630, 4384962, 6641332, 10052272, 15043925, 22501510, 33315580, 49267369, 72250341, 105746966, 153646123

Offset: 0

Stephen DeSalvo, Feb 06 2011, Feb 13 2011

nonn

For two integer partitions of n chosen uniformly at random, a(n)/p(n)^2, where p(n) is the number of partitions of n, is the probability that one dominates the other.
As an example, consider the partitions (4,3,1) and (3,3,2).
4 >= 3, 4+3 >= 3+3, and 4+3+1 = 3+3+2, so we say (4,3,1) majorizes/dominates (3,3,2).
As a non-example, consider (4,1,1,1) and (3,3,1).
4 >= 3, but 4+1 < 3+3, so (4,1,1,1) does NOT dominate (3,3,1).
3 < 4, so (3,3,1) does NOT dominate (4,1,1,1).
Thus the pair (4,1,1,1) and (3,3,1) is not a dominance pair, and does not contribute to a(7).

			For n=1,2,3,4,5, a(n) = p(n)^2, since these values of n give a linear order for integer partitions.

Alois P. Heinz, Table of n, a(n) for n = 0..200 (terms n=1..55 from Stephen DeSalvo)
Wikipedia, Dominance Order
Wikipedia, Majorization

Cf. A000041, A001255, A248476, A265506.

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, min(n-i,i), min(m-j,j), true))))
    end:
a:= n-> 2*b(n$4, true)-combinat[numbpart](n):
seq(a(n), n=0..35);  # Alois P. Heinz, Dec 09 2015

b[n_, m_, i_, j_, t_] := b[n, m, i, j, t] = If[n0, b[n, m, i, j-1, True], 0] + b[n, m, i-1, j, False] + b[n-i, m-j, Min[n-i, i], Min[m-j, j], True]]]]; a[n_] := 2*b[n, n, n, n, True] - PartitionsP[n]; Table[a[n], {n, 0, 35}] (* Jean-François Alcover, Dec 09 2016 after Alois P. Heinz *)

a(0)=1 prepended by Alois P. Heinz, Jul 07 2015

A248475 Number of pairs of partitions of n that are successors in reverse lexicographic order, but incomparable in dominance (natural, majorization) ordering.

Original entry on oeis.org

0, 0, 0, 0, 0, 2, 3, 4, 6, 9, 12, 17, 22, 30, 39, 51, 65, 85, 107, 136, 171, 216, 268, 335, 413, 512, 629, 772, 941, 1151, 1396, 1694, 2046, 2471, 2969, 3569, 4271, 5110, 6093, 7258, 8620, 10235, 12113, 14325, 16902, 19925, 23434, 27540, 32296, 37842, 44260, 51715, 60322, 70306, 81805

Offset: 1

Wouter Meeussen, Oct 07 2014

nonn

Empirical: a(n) is the number of zeros in the subdiagonal of the lower-triangular matrix of coefficients giving the expansion of degree-n complete homogeneous symmetric functions in the Schur basis of the algebra of symmetric functions. - John M. Campbell, Mar 18 2018

			The successor pair (3,1,1,1) and (2,2,2) are incomparable in dominance ordering, and so are their transposes (4,1,1) and (3,3) and these are the two only pairs for n=6, hence a(6)=2.

Ian G. Macdonald, Symmetric functions and Hall polynomials, Oxford University Press, 1979, pp. 6-8.

Wikipedia, Dominance Order

Cf. A182988, A248476, A001475, A076276.

Needs["Combinatorica`"];
dominant[par1_?PartitionQ,par2_?PartitionQ]:= Block[{le=Max[Length[par1],Length[par2]],acc},
acc=Accumulate[PadRight[par1,le]]-Accumulate[PadRight[par2,le]];Which[Min[acc]===0&&Max[acc]>=0,1,Min[acc]<=0&&Max[acc]===0,-1,True,0]];
Table[Count[Apply[dominant, Partition[Partitions[n], 2,1], 1],0], {n,40}]

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.

Original entry on oeis.org

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

Alois P. Heinz, Dec 09 2015

nonn

			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}.

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

Cf. A000009, A000217, A248476, A265506.

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);

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 *)

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

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A182988 The number of dominance pairs of integer partitions of n according to either/or dominance order, where dominance between two partitions x and y means that x is majorized by y or y is majorized by x.

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A248475 Number of pairs of partitions of n that are successors in reverse lexicographic order, but incomparable in dominance (natural, majorization) ordering.

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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.

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