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A326210 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.

%I A326210 #14 Aug 26 2019 12:44:04
%S A326210 0,0,0,0,16,672,29888,2071936,268204288,68717285888,35184350796800,
%T A326210 36028796807919616,73786976292712960000,302231454903635611721728,
%U A326210 2475880078570760326175178752,40564819207303340845566684397568,1329227995784915872903782635437883392
%N A326210 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.
%C A326210 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.
%C A326210 Also simple graphs such that, if the edges are listed in lexicographic order, their maxima (seconds) are not weakly increasing.
%H A326210 Andrew Howroyd, <a href="/A326210/b326210.txt">Table of n, a(n) for n = 0..50</a>
%F A326210 A006125(n) = a(n) + A054726(n).
%e A326210 The a(4) = 16 nesting edge-sets:
%e A326210   {14,23}
%e A326210   {12,14,23}
%e A326210   {13,14,23}
%e A326210   {14,23,24}
%e A326210   {14,23,34}
%e A326210   {12,13,14,23}
%e A326210   {12,14,23,24}
%e A326210   {12,14,23,34}
%e A326210   {13,14,23,24}
%e A326210   {13,14,23,34}
%e A326210   {14,23,24,34}
%e A326210   {12,13,14,23,24}
%e A326210   {12,13,14,23,34}
%e A326210   {12,14,23,24,34}
%e A326210   {13,14,23,24,34}
%e A326210   {12,13,14,23,24,34}
%e A326210 The a(4) = 16 crossing edge-sets:
%e A326210   {13,24}
%e A326210   {12,13,24}
%e A326210   {13,14,24}
%e A326210   {13,23,24}
%e A326210   {13,24,34}
%e A326210   {12,13,14,24}
%e A326210   {12,13,23,24}
%e A326210   {12,13,24,34}
%e A326210   {13,14,23,24}
%e A326210   {13,14,24,34}
%e A326210   {13,23,24,34}
%e A326210   {12,13,14,23,24}
%e A326210   {12,13,14,24,34}
%e A326210   {12,13,23,24,34}
%e A326210   {13,14,23,24,34}
%e A326210   {12,13,14,23,24,34}
%t A326210 Table[Length[Select[Subsets[Subsets[Range[n],{2}]],!OrderedQ[Last/@#]&]],{n,0,5}]
%o A326210 (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
%Y A326210 Non-nesting graphs are A054726.
%Y A326210 Nesting digraphs are A326209.
%Y A326210 Nesting (or crossing) set partitions are A016098.
%Y A326210 MM-numbers of nesting multiset partitions are A326256.
%Y A326210 Cf. A001519, A006125, A095661, A324170, A324172, A324326.
%Y A326210 Cf. A326211, A326243, A326244, A326248, A326250.
%K A326210 nonn
%O A326210 0,5
%A A326210 _Gus Wiseman_, Jun 19 2019
%E A326210 Terms a(7) and beyond from _Andrew Howroyd_, Aug 26 2019