Related papers: Schr\"oder Coloring and Applications
We extend the Marcus-Schaeffer bijection between orientable rooted bipartite quadrangulations (equivalently: rooted maps) and orientable labeled one-face maps to the case of all surfaces, that is orientable and non-orientable as well. This…
Set partitions and permutations with restrictions on the size of the blocks and cycles are important combinatorial sequences. Counting these objects lead to the sequences generalizing the classical Stirling and Bell numbers. The main focus…
A Schr\"oder path is a lattice path from $(0,0)$ to $(2n,0)$ with steps $(1,1)$, $(1,-1)$ and $(2,0)$ that never goes below the $x-$axis. A small Schr\"{o}der path is a Schr\"{o}der path with no $(2,0)$ steps on the $x-$axis. In this paper,…
We revisit a well-known family of polynomial ideals encoding the problem of graph-$k$-colorability. Our paper describes how the inherent combinatorial structure of the ideals implies several interesting algebraic properties. Specifically,…
In this note we augment the poly-Bernoulli family with two new combinatorial objects. We derive formulas for the relatives of the poly-Bernoulli numbers using the appropriate variations of combinatorial interpretations. Our goal is to show…
We define a class of bipartite graphs that correspond naturally with Ferrers diagrams. We give expressions for the number of spanning trees, the number of Hamiltonian paths when applicable, the chromatic polynomial, and the chromatic…
The polynomial ring $B$ in infinitely many indeterminates $(x_1,x_2,\ldots)$, with rational coefficients, has a vector space basis of Schur polynomials, parametrized by partitions. The goal of this note is to provide an explanation of the…
In this note, we provide bijective proofs of some identities involving the Bell number, as previously requested. Our arguments may be extended to yield a generalization in terms of complete Bell polynomials. We also provide a further…
We present a common generalization of counting lattice points in rational polytopes and the enumeration of proper graph colorings, nowhere-zero flows on graphs, magic squares and graphs, antimagic squares and graphs, compositions of an…
Working with generating functions, the combinatorics of a recurrence relation can be expressed in a way that allows for more efficient calculation of the quantity. This is true of the Catalan numbers for an ordered binary tree…
We study decreasing binary trees in which every vertex with two children is colored red or blue. We construct two bijections. The first, to ordered set partitions into odd-sized blocks each arranged as an alternating permutation, shows that…
The Carrell-Chapuy recurrence formulas dramatically improve the efficiency of counting orientable rooted maps by genus, either by number of edges alone or by number of edges and vertices. This paper presents an implementation of these…
In a rather straightforward manner, we develop the well-known formula for the Stirling numbers of the first kind in terms of the (exponential) complete Bell polynomials where the arguments include the generalised harmonic numbers. We also…
The number of inversion sequences avoiding two patterns $101$ and $102$ is known to be the same as the number of permutations avoiding three patterns $2341$, $2431$, and $3241$. This sequence also counts the number of Schr\"{o}der paths…
We introduce a new combinatorial object called tower diagrams and prove fundamental properties of these objects. We also introduce an algorithm that allows us to slide words to tower diagrams. We show that the algorithm is well-defined only…
In this paper we introduce the definition of marked permutations. We first present a bijection between Stirling permutations and marked permutations. We then present an involution on Stirling derangements. Furthermore, we present a…
We define a far-reaching generalization of Schnyder woods which encompasses many classical combinatorial structures on planar graphs. Schnyder woods are defined for planar triangulations as certain triples of spanning trees covering the…
We replace the familiar Stokes vector by a tensor. This allows us to introduce, for example, polar-coordinate components of the Stokes vector. From the tensor we can derive the skyrmion field for mapping the polarization in structured light…
The idea of (combinatorial) Gray codes is to list objects in question in such a way that two successive objects differ in some pre-specified small way. In this paper, we utilize beta-description trees to cyclicly Gray code three classes of…
We study an abstract notion of tree structure which lies at the common core of various tree-like discrete structures commonly used in combinatorics: trees in graphs, order trees, nested subsets of a set, tree-decompositions of graphs and…