A232700 Number of labeled connected point-determining bipartite graphs on n vertices.

1, 1, 0, 12, 60, 1320, 26880, 898800, 40446000, 2568736800, 225962684640, 27627178692960, 4686229692144000, 1104514965434200320, 361988888631722352000, 165271302775469812521600, 105278651889065640047462400, 93750696652129931568573619200

Offset

1,4

Comments

A graph is point-determining if no two vertices have the same set of neighbors. This kind of graph is also called a mating graph.

For all n >= 3, a(n) is even. For every prime p > 2, a(n) is divisible by p for all n >= p. It follows that, if m is squarefree with largest prime factor q, then a(n) is divisible by m for all n >= q. A similar property appears to hold for prime powers, in which case it would hold for all numbers.

Links

Andrew Howroyd, Table of n, a(n) for n = 1..100 (terms 1..20 from Justin M. Troyka)

Ira Gessel and Ji Li, Enumeration of point-determining graphs, arXiv:0705.0042 [math.CO], 2007-2009.

Andy Hardt, Pete McNeely, Tung Phan, and Justin M. Troyka, Combinatorial species and graph enumeration, arXiv:1312.0542 [math.CO], 2013.

Formula

E.g.f.: x + log(1 + (G(x)-x-1)/(1+x)) where G(x) is the e.g.f. of A232699. - Andrew Howroyd, Sep 09 2018

Example

Consider n = 4.  There are 12 connected point-determining bipartite graphs on 4 vertices: the graph *--*--*--*, with 12 possible labelings. - Justin M. Troyka, Nov 27 2013

Mathematica

terms = 18;
G[x_] = Sqrt[Sum[((1 + x)^2^k*Log[1 + x]^k)/k!, {k, 0, terms+1}]] + O[x]^(terms+1);
A[x_] = x + Log[1 + (G[x] - x - 1)/(1 + x)];
(CoefficientList[A[x], x]*Range[0, terms]!) // Rest (* Jean-François Alcover, Sep 13 2018, after Andrew Howroyd *)

Prog

(PARI) seq(n)={my(A=log(1+x+O(x*x^n))); my(p=sqrt(sum(k=0, n, exp(2^k*A)*A^k/k!))); Vec(serlaplace(x + log(1+(p-x-1)/(1+x))))} \\ Andrew Howroyd, Sep 09 2018

Crossrefs

Cf. A006024, A004110 (labeled and unlabeled point-determining graphs).

Cf. A092430, A004108 (labeled and unlabeled connected point-determining graphs).

Cf. A232699, A218090 (labeled and unlabeled point-determining bipartite graphs).

Cf. A088974 (unlabeled connected point-determining bipartite graphs).

Sequence in context: A231026 A333969 A076504 * A354882 A044150 A044531

Adjacent sequences: A232697 A232698 A232699 * A232701 A232702 A232703

Keyword

nonn

Author

Justin M. Troyka, Nov 27 2013

Status

approved