Convex polyominoes revisited: Enumeration of outer site perimeter, interior vertices, and boundary vertices of certain degrees

TL;DR

This paper introduces a new column-by-column decomposition method for convex polyominoes, enabling enumeration of various statistics such as interior vertices, boundary vertices of specific degrees, and outer site perimeter, with detailed asymptotic results.

Contribution

It presents a novel decomposition technique for convex polyominoes that facilitates enumeration of multiple combinatorial statistics and derives their asymptotic behaviors.

Findings

Average interior vertices asymptotic to n^2/12 plus correction terms.

Average boundary vertices of degree two asymptotic to (n+6)/2 plus correction terms.

Number of convex polyominoes with perimeter n asymptotic to a constant times ( (3+√5)/2 )^n.

Abstract

The main contribution of this paper is a new column-by-column method for the decomposition of generating functions of convex polyominoes suitable for enumeration with respect to various statistics including but not limited to interior vertices, boundary vertices of certain degrees, and outer site perimeter. Using this decomposition, among other things, we show that A) the average number of interior vertices over all convex polyominoes of perimeter $2n$ is asymptotic to $\frac{n^2}{12}+\frac{n\sqrt{n}}{3\sqrt{\pi}} -\frac{(21\pi-16)n}{12\pi}.$ B) the average number of boundary vertices with degree two over all convex polyominoes of perimeter $2n$ is asymptotic to $\frac{n+6}{2}+\frac{1}{\sqrt{\pi n}}+\frac{(16-7\pi)}{4\pi n}.$ Additionally, we obtain an explicit generating function counting the number of convex polyominoes with $n$ boundary vertices of degrees at most three and show that this number is asymptotic to $ \frac{n+1}{40}\left(\frac{3+\sqrt{5}}{2}\right)^{n-3} +\frac{\sqrt[4]{5}(2-\sqrt{5})}{80\sqrt{\pi n}}\left(\frac{3+\sqrt{5}}{2}\right)^{n-2}. $ Moreover, we show that the expected number of the boundary vertices of degree four over all convex polyominoes with $n$ vertices of degrees at most three is asymptotically $ \frac{n}{\sqrt{5}}-\frac{\sqrt[4]{125}(\sqrt{5}-1)\sqrt{n}}{10\sqrt{\pi}}. $ C) the number of convex polyominoes with the outer-site perimeter $n$ is asymptotic to $\frac{3(\sqrt{5}-1)}{20\sqrt{\pi n}\sqrt[4]{5}}\left(\frac{3+\sqrt{5}}{2}\right)^n,$ and show the expected number of the outer-site perimeter over all convex polyominoes with perimeter $2n$ is asymptotic to $\frac{25n}{16}+\frac{\sqrt{n}}{4\sqrt{\pi}}+\frac{1}{8}.$ Lastly, we prove that the expected perimeter over all convex polyominoes with the outer-site perimeter $n$ is asymptotic to $\sqrt[4]{5}n$.