Enumeration of Various Animals on the Triangular Lattice

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 $$\frac{1280}{441\sqrt{3\pi n^3}}3^n.$$