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تسجيل مستخدم جديدEnumeration of Various Animals on the Triangular Lattice
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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 $$frac{(xi+1)^2sqrt{xi^4+xi^3-2xi+1}}{2sqrt{pi n^3}}xi^{-n-2},$$ where $xi$ is a root of a certain explicit polynomial of degree 5. (B) column-convex polyiamonds of perimeter $n$ is asymptotic to $$frac{(17997809sqrt{17}+3^3cdot13cdot175463)sqrt{95sqrt{17}-119}}{2^7cdot43^2cdot 89^2sqrt{6pi n^3}}left(frac{3+sqrt{17}}{2}right)^{n-1}.$$ (C) convex polyiamonds of perimeter $n$ is asymptotic to $$frac{1280}{441sqrt{3pi n^3}}3^n.$$
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