Abstract
We introduce a decomposition method for column-convex polyominoes and enumerate them in terms of two statistics: the number of internal vertices and the number of corners in the boundary. We first find the generating function for the column-convex polyominoes according to the horizontal and vertical half-perimeter, and the number of interior vertices. In particular, we show that the average number of interior vertices over all column-convex polyominoes of perimeter 2n is asymptotic to \(\alpha _o n^{3/2}\) where \(\alpha _o\approx 0.57895563\ldots .\) We also find the generating function for the column-convex polyominoes according to the horizontal and vertical half-perimeter, and the number of corners in the boundary. In particular, we show that the average number of corners over all column-convex polyominoes of perimeter 2n is asymptotic to \(\alpha _1n\) where \(\alpha _1\approx 1.17157287\ldots .\)
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Cakić, N., Mansour, T. & Yıldırım, G. A Decomposition of Column-Convex Polyominoes and Two Vertex Statistics. Math.Comput.Sci. 16, 9 (2022). https://doi.org/10.1007/s11786-022-00528-5
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DOI: https://doi.org/10.1007/s11786-022-00528-5