109 research outputs found

### 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

### On the growth rate of 1324-avoiding permutations

We give an improved algorithm for counting the number of $1324$-avoiding
permutations, resulting in 5 further terms of the generating function. We
analyse the known coefficients and find compelling evidence that unlike other
classical length-4 pattern-avoiding permutations, the generating function in
this case does not have an algebraic singularity. Rather, the number of
1324-avoiding permutations of length $n$ behaves as $B\cdot \mu^n \cdot
\mu_1^{n^{\sigma}} \cdot n^g.$ We estimate $\mu=11.60 \pm 0.01,$ $\sigma=1/2,$
$\mu_1 = 0.0398 \pm 0.0010,$ $g = -1.1 \pm 0.2$ and $B =9.5 \pm 1.0.$Comment: 20 pages, 10 figure

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