Families of prudent self-avoiding walks

JOURNAL OF COMBINATORIAL THEORY SERIES A

卷:117

Article

Bousquet-Melou, Mireille

关键词: Enumeration;    Self-avoiding walks;    D-finite series;   

DOI  :  10.1016/j.jcta.2009.10.001

来源: Elsevier

【 摘 要 】

A self-avoiding walk (SAW) oil the square lattice is prudent if it never takes a step towards a vertex it has already visited. Prudent walks differ from most classes of SAW that have been Counted so far in that they call wind around their starting point. Their enumeration was first addressed by Prea in 1997. He defined 4 classes Of prudent walks, of increasing generality, and wrote a system Of recurrence relations for each of them. However, these relations involve more and more parameters as the generality of the class increases. The first class actually consists of partially directed walks, and its generating function is well known to be rational. The second class was proved to have in algebraic (quadratic) generating function by Duchi (2005). Here, we solve exactly the third class, which turns Out to be Much more complex: its generating function is not algebraic, nor even D-finite. The fourth class-general prudent walks-is the only isotropic one, and still defeats Lis. However, we design all isotropic family of prudent walks on the triangular lattice, which We count exactly. Again, the generating function is proved to be non-D-finite. We also Study the asymptotic properties of these classes of walks, with the (somewhat disappointing) conclusion that their endpoint moves away from the origin at a positive speed. This is confirmed visually by the random generation procedures we have designed. (C) 2009 Elsevier Inc. All rights reserved.

【 授权许可】

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