Exact solution of two classes of prudent polygons

$p$ 和 $p^{2}$ 边形的圆度与对称性

Circularity and Symmetries of $p$ and $p^{2}$-polygons

研究具有素数或素数平方个顶点的多边形的对称性，通过对称轴数量分类，并枚举不同对称类别的等价类数目。

Area limit laws for symmetry classes of staircase polygons

We derive area limit laws for the various symmetry classes of staircase polygons on the square lattice, in a uniform ensemble where, for fixed perimeter, each polygon occurs with the same probability. This complements a previous study by Leroux and Rassart, where explicit expressions for the area and perimeter generating functions of these classes have been derived.

Counting triangulations of some classes of subdivided convex polygons

We compute the number of triangulations of a convex $k$-gon each of whose sides is subdivided by $r-1$ points. We find explicit formulas and generating functions, and we determine the asymptotic behaviour of these numbers as $k$ and/or $r$ tend to infinity. We connect these results with the question of finding the planar set of points in general position that has the minimum possible number of triangulations - a well-known open problem from computational geometry.

Kähler-Einstein多边形在突变等价类中的唯一性

On the Uniqueness of Kähler-Einstein Polygons in Mutation-Equivalence Classes

本文研究Kähler-Einstein Fano多边形在突变等价类中的唯一性，证明对称Fano多边形与Kähler-Einstein三角形在突变等价下同构，给出反例并探讨其性质。

Encoding and avoiding 2-connected patterns in polygon dissections and outerplanar graphs

Let $Δ=\{ δ_1,δ_2,...,δ_m \} $ be a finite set of 2-connected patterns, i.e. graphs up to vertex relabelling. We study the generating function $D_{Δ}(z,u_1,u_2,...,u_m),$ which counts polygon dissections and marks subgraph copies of $δ_i$ with the variable $u_i$. We prove that this is always algebraic, through an explicit combinatorial decomposition depending on $Δ$. The decomposition also gives a defining system for $D_{Δ}(z,\mathbf{0})$, which encodes polygon dissections that restrict these patterns as subgraphs. In this way, we are able to extract normal limit laws for the patterns when the…

Neighborhood-Prime Labelings of Trees and Other Classes of Graphs

A neighborhood-prime labeling of a graph is a variation of a prime labeling in which the vertices are assigned labels from $1$ to $|V(G)|$ such that the $\gcd$ of the labels in the neighborhood of each non-degree $1$ vertex is equal to $1$. In this paper, we examine neighborhood-prime labelings of several classes of graphs such as polygonal snakes and books, with a focus on trees including caterpillars, spiders, and firecrackers.

Cellular Structures determined by Polygons and Trees

The polytope structure of the associahedron is decomposed into two categories, types and classes. The classification of types is related to integer partitions, whereas the classes present a new combinatorial problem. We solve this and incorporate the results into properties of the real moduli space of Riemann spheres. Connections are discussed with relation to classic combinatorial problems as well as to other sciences.

On polygon numbers of circle graphs and distance hereditary graphs

Circle graphs are intersection graphs of chords in a circle and $k$-polygon graphs are intersection graphs of chords in a convex $k$-sided polygon where each chord has its endpoints on distinct sides. The $k$-polygon graphs, for $k \ge 2$, form an infinite chain of graph classes, each of which contains the class of permutation graphs. The union of all of those graph classes is the class of circle graphs. The polygon number $\gp(G)$ of a circle graph $G$ is the minimum $k$ such that $G$ is a $k$-polygon graph. Given a circle graph $G$ and an integer $k$, determining whether $\gp(G) \le k$ is NP…

Exact-size Sampling of Enriched Trees in Linear Time

Various combinatorial classes such as outerplanar graphs and maps, series-parallel graphs, substitution-closed classes of permutations and many more allow bijective encodings by so-called enriched trees, which are rooted trees with additional structure on the offspring of each node. Using this universal description we develop sampling procedures that uniformly generate objects from this classes with a given size $n$ in expected time $O(n)$.The key ingredient is a representation of enriched trees in terms of decorated Bienaymé--Galton--Watson trees, which allows us to develop a novel combinatio…

Wythoffian Skeletal Polyhedra in Ordinary Space, I

Skeletal polyhedra are discrete structures made up of finite, flat or skew, or infinite, helical or zigzag, polygons as faces, with two faces on each edge and a circular vertex-figure at each vertex. When a variant of Wythoff's construction is applied to the forty-eight regular skeletal polyhedra (Grunbaum-Dress polyhedra) in ordinary space, new highly symmetric skeletal polyhedra arise as "truncations" of the original polyhedra. These Wythoffians are vertex-transitive and often feature vertex configurations with an attractive mix of different face shapes. The present paper describes the bluep…