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 are asymptotically [Formula presented]ξ−n−2,where ξ is a root of a certain explicit polynomial of degree 5. (B) column-convex polyiamonds of perimeter n are asymptotic to [Formula presented][Formula presented]n−1. (C) convex polyiamonds of perimeter n is asymptotic to [Formula presented]3n.
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ASJC Scopus subject areas
- Discrete Mathematics and Combinatorics