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A182988
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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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5
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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
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OFFSET
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0,3
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COMMENTS
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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).
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LINKS
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EXAMPLE
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For n=1,2,3,4,5, a(n) = p(n)^2, since these values of n give a linear order for integer partitions.
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MAPLE
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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, min(n-i, i), min(m-j, j), true))))
end:
a:= n-> 2*b(n$4, true)-combinat[numbpart](n):
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MATHEMATICA
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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, 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 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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