论文标题
重新审视的凸多粒子：外部位置周长，内部顶点和一定程度的边界顶点的枚举
Convex polyominoes revisited: Enumeration of outer site perimeter, interior vertices, and boundary vertices of certain degrees
论文作者
论文摘要
本文的主要贡献是一种新的列列方法，用于生成适合枚举的凸多个群落功能的分解，适用于各种统计数据，包括但不限于内部顶点，某些程度的边界顶点和外部位点的边界顶点。使用这种分解，除其他外，我们表明 a）所有凸出的凸出$ 2N $的内部顶点的平均数量是渐近的，渐近至$ \ frac {n^2} {12}+\ frac {n \ sqrt {n}}}} b）在所有凸出$ 2n $的凸多个群体中，具有二维的边界顶点的平均数量渐近至$ \ frac {n+6} {2} {2}+\ frac {1} {\ sqrt {\ sqrt {πn}}}}}}}}}+\ frac {（16-7π）最多三分的凸doloMinos具有$ n $界限的数量，并表明该数字是渐近的，对$ \ frac {n+1} {40} {40} \ left（\ frac {3+ \ sqrt {3+ \ sqrt {5}}}}}} {2} {2} {2} {2} {2} {n-3} +\ frac {\ sqrt [4] {5}（2- \ sqrt {5}）}} {80 \ sqrt {πn}} \ left（\ frac {3+ \ sqrt {5}}}}} {2} {2} {2} {2} \ right） $此外，我们表明，在所有凸多个群体中，最多三个学位的所有凸多个粉丝的预期边界顶点的数量最多是三个。 \ frac {n} {\ sqrt {5}} - \ frac {\ sqrt [4] {125}（\ sqrt {5} -1）\ sqrt {n}}} {10 \sqrtπ}。 $ c）带有外部位点$ n $的凸多元的数量是渐近的，均为$ \ frac {3（\ sqrt {5} -1）} {20 \ sqrt {π n} \ sqrt [4] {5}} \ left（\ frac {3+ \ sqrt {5}}} {2}} {2} {2} \ right）^n，$，并显示所有convex polyominoes to Emptotic to asmptotic to asmptotic to asymptotic to asymptotic to asymptotic to asmptotic to asymptotic to $ \ frac {25n} {16}+\ frac {\ sqrt {\ sqrt {n}} {4 \sqrtπ}+\ frac {1} {1} {8} $最后，我们证明，我们证明了所有预期的外观范围内的凸范围多粒元素$ n $ $ n $ nisempt osympt asmpt asmpt asmpt asmpt asmpt asmptt $ \ sqrt [4] {5} n $。
The main contribution of this paper is a new column-by-column method for the decomposition of generating functions of convex polyominoes suitable for enumeration with respect to various statistics including but not limited to interior vertices, boundary vertices of certain degrees, and outer site perimeter. Using this decomposition, among other things, we show that A) the average number of interior vertices over all convex polyominoes of perimeter $2n$ is asymptotic to $\frac{n^2}{12}+\frac{n\sqrt{n}}{3\sqrtπ} -\frac{(21π-16)n}{12π}.$ B) the average number of boundary vertices with degree two over all convex polyominoes of perimeter $2n$ is asymptotic to $\frac{n+6}{2}+\frac{1}{\sqrt{πn}}+\frac{(16-7π)}{4πn}.$ Additionally, we obtain an explicit generating function counting the number of convex polyominoes with $n$ boundary vertices of degrees at most three and show that this number is asymptotic to $ \frac{n+1}{40}\left(\frac{3+\sqrt{5}}{2}\right)^{n-3} +\frac{\sqrt[4]{5}(2-\sqrt{5})}{80\sqrt{πn}}\left(\frac{3+\sqrt{5}}{2}\right)^{n-2}. $ Moreover, we show that the expected number of the boundary vertices of degree four over all convex polyominoes with $n$ vertices of degrees at most three is asymptotically $ \frac{n}{\sqrt{5}}-\frac{\sqrt[4]{125}(\sqrt{5}-1)\sqrt{n}}{10\sqrtπ}. $ C) the number of convex polyominoes with the outer-site perimeter $n$ is asymptotic to $\frac{3(\sqrt{5}-1)}{20\sqrt{π n}\sqrt[4]{5}}\left(\frac{3+\sqrt{5}}{2}\right)^n,$ and show the expected number of the outer-site perimeter over all convex polyominoes with perimeter $2n$ is asymptotic to $\frac{25n}{16}+\frac{\sqrt{n}}{4\sqrtπ}+\frac{1}{8}.$ Lastly, we prove that the expected perimeter over all convex polyominoes with the outer-site perimeter $n$ is asymptotic to $\sqrt[4]{5}n$.
