论文标题
在三角形晶格上列举各种动物
Enumeration of Various Animals on the Triangular Lattice
论文作者
论文摘要
在本文中，我们考虑了位于三角形晶格上的各种聚原木。通过仔细分析通过一层分解和细胞修剪/成长的参数，我们为非空翻译不变的baryiamonds的生成函数（三角形lattice中的条形图），柱 - 凸线Polyiamonds和Convex Polyiamonds的生成函数提供了明确的形式。特别是，我们证明了 （a）周长$ n $的baryiamonds在$$上是$$ \ frac {（（ξ+1）^2 \ sqrt {ξ^4+ξ^4+ξ^3-2 ex+1}} {2 \ sqrt {πn^3}}} explial $ 5 protem的$ 5^^^^^^^^{ - n-2} （b）列周围$ n $的列convex polyiamonds渐近\ frac {（17997809 \ sqrt {17}+3^3^3 \ cdot13 \ cdot175463 \ cdot175463） 89^2 \ sqrt {6πn^3}}} \ left（\ frac {3+ \ sqrt {17}}} {2} {2} \ right）^{n-1}。$$ （c）外围$ n $的凸polyiamonds渐近\ $ \ frac {1280} {441 \ sqrt {3πn^3}} 3^n。$$
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 $$\frac{(ξ+1)^2\sqrt{ξ^4+ξ^3-2ξ+1}}{2\sqrt{πn^3}}ξ^{-n-2},$$ where $ξ$ is a root of a certain explicit polynomial of degree 5. (B) column-convex polyiamonds of perimeter $n$ is asymptotic to $$\frac{(17997809\sqrt{17}+3^3\cdot13\cdot175463)\sqrt{95\sqrt{17}-119}}{2^7\cdot43^2\cdot 89^2\sqrt{6πn^3}}\left(\frac{3+\sqrt{17}}{2}\right)^{n-1}.$$ (C) convex polyiamonds of perimeter $n$ is asymptotic to $$\frac{1280}{441\sqrt{3πn^3}}3^n.$$
