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 is asymptotic to (B) the average number of boundary vertices with degree two over all convex polyominoes of perimeter is asymptotic to 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 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 (C) the number of convex polyominoes with the outer-site perimeter n is asymptotic to and show the expected number of the outer-site perimeter over all convex polyominoes with perimeter is asymptotic to Lastly, we prove that the expected perimeter over all convex polyominoes with the outer-site perimeter n is asymptotic to .

中文翻译：

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凸多联骨牌重温：外部场地周长、内部顶点和特定度数的边界顶点的枚举

本文的主要贡献是一种新的逐列分解凸多联式的生成函数的方法，适用于各种统计的枚举，包括但不限于内部顶点、一定度的边界顶点和外部站点周长. 使用这种分解，除其他外，我们表明 (A) 周长所有凸多联式上的平均内部顶点数渐近于 (B) 周长所有凸多联式上的度数为 2 的边界顶点的平均数渐近于此外，我们获得了一个显式的生成函数，该函数计算具有 n 个度数最多为 3 的边界顶点的凸多联式的数量，并表明这个数字是渐近的，此外，