Enumeration of Various Animals on the Triangular Lattice

TL;DR

This paper derives explicit generating functions and asymptotic formulas for counting various classes of polyiamonds on the triangular lattice, including baryiamonds, column-convex, and convex polyiamonds, using layer decompositions.

Contribution

It provides new explicit generating functions and asymptotic counts for several classes of polyiamonds on the triangular lattice, advancing combinatorial enumeration methods.

Findings

Asymptotic count for baryiamonds involves a root of a degree 5 polynomial.

Explicit asymptotic formulas for column-convex polyiamonds.

Asymptotic enumeration of convex polyiamonds with a simple exponential growth rate.

Abstract

In this paper, we consider various classes of polyiamonds that are animals residing on the triangular lattice. By careful analyses through certain layer-by-layer decompositions and cell pruning/growing arguments, we derive explicit forms for the generating functions of the number of nonempty translation-invariant baryiamonds (bargraphs in the triangular lattice), column-convex polyiamonds, and convex polyiamonds with respect to their perimeter. In particular, we show that the number of (A) baryiamonds of perimeter $n$ is asymptotically $$\frac{(\xi+1)^2\sqrt{\xi^4+\xi^3-2\xi+1}}{2\sqrt{\pi n^3}}\xi^{-n-2},$$ where $\xi$ is a root of a certain explicit polynomial of degree 5. (B) column-convex polyiamonds of perimeter $n$ is asymptotic to $$\frac{(17997809\sqrt{17}+3^3\cdot13\cdot175463)\sqrt{95\sqrt{17}-119}}{2^7\cdot43^2\cdot 89^2\sqrt{6\pi n^3}}\left(\frac{3+\sqrt{17}}{2}\right)^{n-1}.$$ (C) convex polyiamonds of perimeter $n$ is asymptotic to $$\frac{1280}{441\sqrt{3\pi n^3}}3^n.$$