A221493 Number of tangled bicolored graphs on n labeled vertices.

0, 0, 0, 0, 12, 120, 2460, 64680, 2323692, 111920760, 7272700860, 639739653960, 76764606923532, 12645557866982040, 2878366780307114460, 909775941009828296040, 401039212596034472197932, 247339947733328456032703160, 214013123181627427780427544060

Offset

0,5

Comments

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

[ A | J ]

[---+---]

[ 0 | B ]

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

Links

Table of n, a(n) for n=0..18.

M. Guay-Paquet, A. H. Morales, E. Rowland, Structure and enumeration of (3+1)-free posets (extended abstract), arXiv:1212.5356 [math.CO], 2012.

Formula

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

Example

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.

Mathematica

nmax = 19;
B[x_] = Sum[Exp[2^n x] x^n/n!, {n, 0, nmax}] + O[x]^nmax;
T[x_] = 2 Exp[-x] - 1 - 1/B[x] + O[x]^nmax;
CoefficientList[T[x], x] Range[0, nmax-1]! (* Jean-François Alcover, Aug 12 2018 *)

Crossrefs

Cf. A221492, A079145.

Sequence in context: A102341 A174561 A009078 * A009149 A010570 A012374

Adjacent sequences: A221490 A221491 A221492 * A221494 A221495 A221496

Keyword

nonn,easy

Author

Mathieu Guay-Paquet, Jan 18 2013

Status

approved