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

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

In the collections

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].

## Links

## Other information

- issn
- 00973165
- journal
- Journal of Combinatorial Theory, Series A
- keywords
- Bond animals,Differentiably finite power series,Enumeration,Solvability
- number
- 6
- pages
- 1031--1049
- volume
- 113

### BibTeX entry

@article{Rechnitzer2006, title = {Haruspicy 3: The anisotropic generating function of directed bond-animals is not D-finite}, author = {Rechnitzer, Andrew}, url = {http://www.sciencedirect.com/science/article/pii/S0097316505001883}, urldate = {2015-11-08}, 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 = {}, 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, collections = {Animals,Attention-grabbing titles} }