A232700 - OEIS

%I #26 Sep 13 2018 13:20:51

%S 1,1,0,12,60,1320,26880,898800,40446000,2568736800,225962684640,

%T 27627178692960,4686229692144000,1104514965434200320,

%U 361988888631722352000,165271302775469812521600,105278651889065640047462400,93750696652129931568573619200

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

%C 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.

%C 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.

%H Andrew Howroyd, <a href="/A232700/b232700.txt">Table of n, a(n) for n = 1..100</a> (terms 1..20 from Justin M. Troyka)

%H Ira Gessel and Ji Li, <a href="http://arxiv.org/abs/0705.0042">Enumeration of point-determining graphs</a>, arXiv:0705.0042 [math.CO], 2007-2009.

%H Andy Hardt, Pete McNeely, Tung Phan, and Justin M. Troyka, <a href="http://arxiv.org/abs/1312.0542">Combinatorial species and graph enumeration</a>, arXiv:1312.0542 [math.CO], 2013.

%F 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

%e 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

%t terms = 18;

%t G[x_] = Sqrt[Sum[((1 + x)^2^k*Log[1 + x]^k)/k!, {k, 0, terms+1}]] + O[x]^(terms+1);

%t A[x_] = x + Log[1 + (G[x] - x - 1)/(1 + x)];

%t (CoefficientList[A[x], x]*Range[0, terms]!) // Rest (* _Jean-François Alcover_, Sep 13 2018, after _Andrew Howroyd_ *)

%o (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

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

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

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

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

%K nonn

%O 1,4

%A _Justin M. Troyka_, Nov 27 2013