Haruspicy 3: The anisotropic generating function of directed bond-animals is not D-finite

• Published in 2006
• Added on 2015-11-08

While directed site-animals have been solved on several lattices, directed bond-animals remain unsolved on any nontrivial lattice. In this paper we demonstrate that the anisotropic generating function of directed bond-animals on the square lattice is fundamentally different from that of directed site-animals in that it is not differentiably finite. We also extend this result to directed bond-animals on hypercubic lattices. This indicates that directed bond-animals are unlikely to be solved by similar methods to those used in the solution of directed site-animals. It also implies that a solution cannot be conjectured using computer packages such as Gfun [A Maple package developed by B. Salvy, P. Zimmermann, E. Murray at INRIA, France, available from http://algo.inria.fr/libraries/ at time of submission; B. Salvy, P. Zimmermann, Gfun: A Maple package for the manipulation of generating and holonomic functions in one variable, ACM Trans. Math. Software 20 (2) (1994) 163–177] or differential approximants [A.J. Guttmann, Asymptotic analysis of coefficients, in: C. Domb, J. Lebowitz (Eds.), Phase Transit. Crit. Phenom., vol. 13, Academic Press, London, 1989, pp. 1–234, programs available from http://www.ms.unimelb.edu.au/~tonyg].

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BibTeX entry

@article{Rechnitzer2006,
	key = {Rechnitzer2006},
	type = {article},
	title = {Haruspicy 3: The anisotropic generating function of directed bond-animals is not D-finite},
	author = {Rechnitzer, Andrew},
	abstract = {While directed site-animals have been solved on several lattices, directed bond-animals remain unsolved on any nontrivial lattice. In this paper we demonstrate that the anisotropic generating function of directed bond-animals on the square lattice is fundamentally different from that of directed site-animals in that it is not differentiably finite. We also extend this result to directed bond-animals on hypercubic lattices. This indicates that directed bond-animals are unlikely to be solved by similar methods to those used in the solution of directed site-animals. It also implies that a solution cannot be conjectured using computer packages such as Gfun [A Maple package developed by B. Salvy, P. Zimmermann, E. Murray at INRIA, France, available from http://algo.inria.fr/libraries/ at time of submission; B. Salvy, P. Zimmermann, Gfun: A Maple package for the manipulation of generating and holonomic functions in one variable, ACM Trans. Math. Software 20 (2) (1994) 163–177] or differential approximants [A.J. Guttmann, Asymptotic analysis of coefficients, in: C. Domb, J. Lebowitz (Eds.), Phase Transit. Crit. Phenom., vol. 13, Academic Press, London, 1989, pp. 1–234, programs available from http://www.ms.unimelb.edu.au/{\~{}}tonyg].},
	comment = {},
	date_added = {2015-11-08},
	date_published = {2006-08-01},
	urls = {http://www.sciencedirect.com/science/article/pii/S0097316505001883},
	collections = {Animals,Attention-grabbing titles},
	url = {http://www.sciencedirect.com/science/article/pii/S0097316505001883},
	urldate = {2015-11-08},
	issn = 00973165,
	journal = {Journal of Combinatorial Theory, Series A},
	keywords = {Bond animals,Differentiably finite power series,Enumeration,Solvability},
	month = {aug},
	number = 6,
	pages = {1031--1049},
	volume = 113,
	year = 2006
}