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This article presents unified bijective constructions for planar maps, with control on the face degrees and on the girth. Recall that the girth is the length of the smallest cycle, so that maps of girth at least $d=1,2,3$ are respectively…
In this paper, we enumerate two families of polycubes, the directed plateau polycubes and the plateau polycubes, with respect to the width and a new parameter, the Lateral Area. We give an explicit formula and the generating function for…
Discrete orthogonal matrices have several applications in information technology, such as in coding and cryptography. It is often challenging to generate discrete orthogonal matrices. A common approach widely in use is to discretize…
Balinski (1961) proved that the graph of a $d$-dimensional convex polytope is $d$-connected. We provide a new proof of this result. Our proof provides details on the nature of a separating set with exactly $d$ vertices; some of which appear…
D-finite functions and P-recursive sequences are defined in terms of linear differential and recurrence equations with polynomial coefficients. In this paper, we introduce a class of numbers closely related to D-finite functions and…
Lin and Chang gave a generating function of convex polyominoes with an $m+1$ by $n+1$ minimal bounding rectangle. Gessel showed that their result implies that the number of such polyominoes is $$ \frac{m+n+mn}{m+n}{2m+2n\choose…
Given a digraph $D=(V,A)$ on $n$ vertices and a vertex $v\in V$, the cycle-degree of $v$ is the minimum size of a set $S \subseteq V(D) \setminus \{v\}$ intersecting every directed cycle of $D$ containing $v$. From this definition of…
A set $X$ of vertices of an acyclic digraph $D$ is convex if $X\neq \emptyset$ and there is no directed path between vertices of $X$ which contains a vertex not in $X$. A set $X$ is connected if $X\neq \emptyset$ and the underlying…
Due to their broad application to different fields of theory and practice, generalized Petersen graphs $GPG(n,s)$ have been extensively investigated. Despite the regularity of generalized Petersen graphs, determining an exact formula for…
Enumeration of various types of lattice polygons and in particular polyominoes is of primary importance in many machine learning, pattern recognition, and geometric analysis problems. In this work, we develop a large deviation principle for…
Given a weighted undirected graph $G=(V,E,d)$, the Molecular Distance Geometry Problem (MDGP) is that of finding a function $x:G\to \mathbb{R}^{3}$, where $||x(u)-x(v)||=d(u,v)$ for each $\{u,v\}\in E$. We show that under a few assumptions…
We consider the problem of decomposing the edges of a directed graph into as few paths as possible. There is a natural lower bound for the number of paths needed in an edge decomposition of a directed graph $D$ in terms of its degree…
Let $D$ be a strongly connected oriented graph with vertex-set $V$ and arc-set $A$. The distance from a vertex $u$ to another vertex $v$, $d(u,v)$ is the minimum length of oriented paths from $u$ to $v$. Suppose $B=\{b_1,b_2,b_3,...b_k\}$…
In this paper, we propose DDPM-Polycube, a generative polycube creation approach based on denoising diffusion probabilistic models (DDPM) for generating high-quality hexahedral (hex) meshes and constructing volumetric splines. Unlike…
The critical exponents for a class of one-dimensional models of interface depinning in disordered media can be calculated through a mapping onto directed percolation (DP). In higher dimensions these models give rise to directed surfaces,…
We derive self-reciprocity properties for a number of polyomino generating functions, including several families of column-convex polygons, three-choice polygons and staircase polygons with a staircase hole. In so doing, we establish a…
Bipolar orientations of planar maps have recently attracted some interest in combinatorics, probability theory and theoretical physics. Plane bipolar orientations with $n$ edges are known to be counted by the $n$th Baxter number $b(n)$,…
We study the adsorption-desorption phase transition of directed branched polymer in $d+1$ dimensions in contact with a line by mapping it to a $d$ dimensional hard core lattice gas at negative activity. We solve the model exactly in 1+1…
A stacking operation adds a $d$-simplex on top of a facet of a simplicial $d$-polytope while maintaining the convexity of the polytope. A stacked $d$-polytope is a polytope that is obtained from a $d$-simplex and a series of stacking…
The main contribution of this paper is to find a representation of the class $\mathcal{F}_d(p)$ of multivariate Bernoulli distributions with the same mean $p$ that allows us to find its generators analytically in any dimension. We map…