A Decomposition of Column-Convex Polyominoes and Two Vertex Statistics

Nenad Cakić; Toufik Mansour; Gökhan Yıldırım

doi:10.1007/s11786-022-00528-5

A Decomposition of Column-Convex Polyominoes and Two Vertex Statistics

Nenad Cakić, Toufik Mansour, Gökhan Yıldırım

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

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 α_on^{3 / 2} where α_o≈ 0.57895563 …. 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 α₁n where α₁≈ 1.17157287 ….

Original language	English
Article number	9
Journal	Mathematics in Computer Science
Volume	16
Issue number	1
DOIs	https://doi.org/10.1007/s11786-022-00528-5
State	Published - Mar 2022

Bibliographical note

Publisher Copyright:
© 2022, The Author(s), under exclusive licence to Springer Nature Switzerland AG.

Keywords

Bondary vertices
Interior vertices
Kernel method
Polyominoes

ASJC Scopus subject areas

Computational Mathematics
Computational Theory and Mathematics
Applied Mathematics

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10.1007/s11786-022-00528-5

Cite this

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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 αon3 / 2 where αo≈ 0.57895563 …. 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 α1n where α1≈ 1.17157287 ….",

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AU - Yıldırım, Gökhan

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Y1 - 2022/3

N2 - 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 αon3 / 2 where αo≈ 0.57895563 …. 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 α1n where α1≈ 1.17157287 ….

AB - 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 αon3 / 2 where αo≈ 0.57895563 …. 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 α1n where α1≈ 1.17157287 ….

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A Decomposition of Column-Convex Polyominoes and Two Vertex Statistics

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

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