A221493 - OEIS

%I #16 Apr 07 2020 21:33:33

%S 0,0,0,0,12,120,2460,64680,2323692,111920760,7272700860,639739653960,

%T 76764606923532,12645557866982040,2878366780307114460,

%U 909775941009828296040,401039212596034472197932,247339947733328456032703160,214013123181627427780427544060

%N Number of tangled bicolored graphs on n labeled vertices.

%C A bicolored graph on n labeled vertices, k of which are black, and (n-k) of which are white, can be represented as a k X (n-k) matrix, where the (i,j) entry is 1 if the i-th black vertex is adjacent to the j-th white vertex, and 0 otherwise. Then, the graph is tangled if (1) the matrix does not have any rows or columns of all 0's or all 1's; and (2) it is not possible to permute the rows of the matrix and the columns of the matrix to obtain a matrix of the form

%C [ A | J ]

%C [---+---]

%C [ 0 | B ]

%C where the top right block J consists of all 1's, and the bottom left block 0 consists of all 0's.

%H M. Guay-Paquet, A. H. Morales, E. Rowland, <a href="http://arxiv.org/abs/1212.5356">Structure and enumeration of (3+1)-free posets (extended abstract)</a>, arXiv:1212.5356 [math.CO], 2012.

%F E.g.f.: T(x) = 2*e^(-x) - 1 - 1/B(x), where B(x) is the e.g.f. for A047863.

%e The only tangled bicolored graph on 4 vertices (up to isomorphism) consists of 2 black vertices, 2 white vertices, and 2 edges, with each black vertex joined to a different white vertex. Given 4 labels, there are 12 distinct ways of labeling the vertices, so a(4) = 1.

%t nmax = 19;

%t B[x_] = Sum[Exp[2^n x] x^n/n!, {n, 0, nmax}] + O[x]^nmax;

%t T[x_] = 2 Exp[-x] - 1 - 1/B[x] + O[x]^nmax;

%t CoefficientList[T[x], x] Range[0, nmax-1]! (* _Jean-François Alcover_, Aug 12 2018 *)

%Y Cf. A221492, A079145.

%K nonn,easy

%O 0,5

%A _Mathieu Guay-Paquet_, Jan 18 2013