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A326210
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Number of labeled simple graphs with vertices {1..n} containing a nesting pair of edges, where two edges {a,b}, {c,d} are nesting if a < c and b > d or a > c and b < d.
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20
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0, 0, 0, 0, 16, 672, 29888, 2071936, 268204288, 68717285888, 35184350796800, 36028796807919616, 73786976292712960000, 302231454903635611721728, 2475880078570760326175178752, 40564819207303340845566684397568, 1329227995784915872903782635437883392
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OFFSET
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0,5
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COMMENTS
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Also simple graphs containing a crossing pair of edges, where two edges {a,b}, {c,d} are crossing if a < c < b < d or c < a < d < b.
Also simple graphs such that, if the edges are listed in lexicographic order, their maxima (seconds) are not weakly increasing.
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LINKS
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FORMULA
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EXAMPLE
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The a(4) = 16 nesting edge-sets:
{14,23}
{12,14,23}
{13,14,23}
{14,23,24}
{14,23,34}
{12,13,14,23}
{12,14,23,24}
{12,14,23,34}
{13,14,23,24}
{13,14,23,34}
{14,23,24,34}
{12,13,14,23,24}
{12,13,14,23,34}
{12,14,23,24,34}
{13,14,23,24,34}
{12,13,14,23,24,34}
The a(4) = 16 crossing edge-sets:
{13,24}
{12,13,24}
{13,14,24}
{13,23,24}
{13,24,34}
{12,13,14,24}
{12,13,23,24}
{12,13,24,34}
{13,14,23,24}
{13,14,24,34}
{13,23,24,34}
{12,13,14,23,24}
{12,13,14,24,34}
{12,13,23,24,34}
{13,14,23,24,34}
{12,13,14,23,24,34}
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MATHEMATICA
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Table[Length[Select[Subsets[Subsets[Range[n], {2}]], !OrderedQ[Last/@#]&]], {n, 0, 5}]
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PROG
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(PARI) seq(n)={my(p=1 + 3/2*x - x^2 - x/2*sqrt(1 - 12*x + 4*x^2 + O(x^n))); concat([0], vector(n, k, 2^binomial(k, 2)-polcoef(p, k)))} \\ Andrew Howroyd, Aug 26 2019
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CROSSREFS
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Nesting (or crossing) set partitions are A016098.
MM-numbers of nesting multiset partitions are A326256.
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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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