Self-Avoiding Walks and Fibonacci Numbers
نویسندگان
چکیده
By combinatorial arguments, we prove that the number of self-avoiding walks on the strip {0, 1} × Z is 8Fn − 4 when n is odd and is 8Fn − n when n is even. Also, when backwards moves are prohibited, we derive simple expressions for the number of length n self-avoiding walks on {0, 1} × Z, Z× Z, the triangular lattice, and the cubic lattice.
منابع مشابه
Self-avoiding walks, the language of science, and Fibonacci numbers
The self-avoiding walk, restricted to a strip, is considered in the context of linguistic combinatorics. AMS class!Jications: 05A15; 05E15; 60J15
متن کاملSelf-avoiding walks and polygons on quasiperiodic tilings
We enumerate self-avoiding walks and polygons, counted by perimeter, on the quasiperiodic rhombic Penrose and Ammann-Beenker tilings, thereby considerably extending previous results. In contrast to similar problems on regular lattices, these numbers depend on the chosen initial vertex. We compare different ways of counting and demonstrate that suitable averaging improves convergence to the asym...
متن کاملScaling of the atmosphere of self-avoiding walks
Abstract The number of free sites next to the end of a self-avoiding walk is known as the atmosphere of the walk. The average atmosphere can be related to the number of configurations. Here we study the distribution of atmospheres as a function of length and how the number of walks of fixed atmosphere scale. Certain bounds on these numbers can be proved. We use Monte Carlo estimates to verify o...
متن کاملScaling of Self-Avoiding Walks and Self-Avoiding Trails in Three Dimensions
Motivated by recent claims of a proof that the length scale exponent for the end-to-end distance scaling of self-avoiding walks is precisely 7/12 = 0.5833 . . ., we present results of large-scale simulations of self-avoiding walks and self-avoiding trails with repulsive contact interactions on the hypercubic lattice. We find no evidence to support this claim; our estimate ν = 0.5874(2) is in ac...
متن کاملSet partition statistics and q-Fibonacci numbers
We consider the set partition statistics ls and rb introduced by Wachs and White and investigate their distribution over set partitions avoiding certain patterns. In particular, we consider those set partitions avoiding the pattern 13/2, Πn(13/2), and those avoiding both 13/2 and 123, Πn(13/2, 123). We show that the distribution over Πn(13/2) enumerates certain integer partitions, and the distr...
متن کامل