无向对角凸多项式的周长generating function
The perimeter generating function for nondirected diagonally convex polyominoes
摘要：
一个polyomino是一个有限的，边连通集的细胞在平面上。目前，还看不到所有多角形的枚举。另一方面，有几个子集的polyominoes的生成函数已知。例如，存在关于列凸多项式的广泛知识，这是Temperley在1956年引入的模型。在研究列凸多角形时，研究人员也研究了对角凸多角形（DCP），但注意到一个尴尬的功能：当DCP的最后一个对角线被删除时，剩下的对象并不总是多角形。因此，研究人员将注意力集中在定向DCP上。（有向DCP是这样一种DCP，即对于任何$i$，当其最后的$i$对角线被删除时，它仍然是一个多角形。）定向DCP逐渐得到很好的理解，而一般DCP至今仍未被探索。在本文中，我们最终面对一般DCP。模一个小技巧，这使我们免于处理非polyominoes，我们使用分层的方法（描述在书的第3章“Polyominoes，Polyominoes和Polycubes”，由Anthony Guttmann编辑）。这些计算量大得惊人.我们的主要结果是DCP的周长生成函数，我们表示为$D(d,x)$。函数$D(d,x)$是代数的，并且满足八次方程。$D(d,x)$的公式大约有八页长。该公式涉及$d$和$x$中的九个多项式，并且这些多项式中的每一个在$x$中具有$58$或更高的次数。感兴趣的读者可以在本文所附的Maple工作表中查看$D(d,x)$的公式。
英文摘要：
A polyomino is a finite, edge-connected set of cells in the plane. At the present time, an enumeration of all polyominoes is nowhere in sight. On the other hand, there are several subsets of polyominoes for which generating functions are known. For example, there exists extensive knowledge about column-convex polyominoes, a model introduced by Temperley in 1956. While studying column-convex polyominoes, researchers also gave a look at diagonally convex polyominoes (DCPs), but noticed an awkward feature: when the last diagonal of a DCP is deleted, the remaining object is not always a polyomino. So researchers focused their attention on directed DCPs. (A directed DCP is such a DCP that remains a polyomino when, for any $i$, its last $i$ diagonals are deleted.) Directed DCPs gradually became well understood, whereas general DCPs have remained unexplored up to now. In this paper, we finally face general DCPs. Modulo a little trick, which saves us from dealing with non-polyominoes, we use the layered approach (described in chapter 3 of the book ``Polygons, Polyominoes and Polycubes", edited by Anthony Guttmann). The computations are of remarkable bulk. Our main result is the perimeter generating function for DCPs; we denote it $D(d,x)$. The function $D(d,x)$ is algebraic and satisfies an equation of degree eight. The formula for $D(d,x)$ is about eight pages long. That formula involves nine polynomials in $d$ and $x$, and each of those polynomials is of degree $58$ or more in $x$. The interested reader can view the formula for $D(d,x)$ in the Maple worksheet attached to this paper.
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