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Abstract: Column-convex polyominoes were introduced in 1950-s by Temperley, amathematical physicist working on -lattice gases-. By now, column-convexpolyominoes are a popular and well-understood model. There exist severalgeneralizations of column-convex polyominoes; an example is a model calledmulti-directed animals. In this paper, we introduce a new sequence of supersetsof column-convex polyominoes. Our model we call it level m column-subconvexpolyominoes is defined in a simple way. We focus on the case when cells arehexagons and we compute the area generating functions for the levels one andtwo. Both of those generating functions are complicated q-series, whereas thearea generating function of column-convex polyominoes is a rational function.The growth constants of level one and level two column-subconvex polyominoesare 4.319139 and 4.509480, respectively. For comparison, the growth constantsof column-convex polyominoes, multi-directed animals and all polyominoes are3.863131, 4.587894 and 5.183148, respectively.



Author: Svjetlan Feretic, Anthony J. Guttmann

Source: https://arxiv.org/



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