2,220 research outputs found
Enumeration of self avoiding trails on a square lattice using a transfer matrix technique
We describe a new algebraic technique, utilising transfer matrices, for
enumerating self-avoiding lattice trails on the square lattice. We have
enumerated trails to 31 steps, and find increased evidence that trails are in
the self-avoiding walk universality class. Assuming that trails behave like , we find and .Comment: To be published in J. Phys. A:Math Gen. Pages: 16 Format: RevTe
Polyominoes with nearly convex columns: An undirected model
Column-convex polyominoes were introduced in 1950's by Temperley, a
mathematical physicist working on "lattice gases". By now, column-convex
polyominoes are a popular and well-understood model. There exist several
generalizations of column-convex polyominoes; an example is a model called
multi-directed animals. In this paper, we introduce a new sequence of supersets
of column-convex polyominoes. Our model (we call it level m column-subconvex
polyominoes) is defined in a simple way. We focus on the case when cells are
hexagons and we compute the area generating functions for the levels one and
two. Both of those generating functions are complicated q-series, whereas the
area generating function of column-convex polyominoes is a rational function.
The growth constants of level one and level two column-subconvex polyominoes
are 4.319139 and 4.509480, respectively. For comparison, the growth constants
of column-convex polyominoes, multi-directed animals and all polyominoes are
3.863131, 4.587894 and 5.183148, respectively.Comment: 26 pages, 14 figure
Fuchsian differential equation for the perimeter generating function of three-choice polygons
Using a simple transfer matrix approach we have derived very long series
expansions for the perimeter generating function of three-choice polygons. We
find that all the terms in the generating function can be reproduced from a
linear Fuchsian differential equation of order 8. We perform an analysis of the
properties of the differential equation.Comment: 13 pages, 2 figures, talk presented in honour of X. Viennot at
Seminaire Lotharengien, Lucelle, France, April 3-6 2005. Paper amended and
sligtly expanded after refereein
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