Related papers: Eulerian digraphs and Dyck words, a bijection
A connected digraph in which the in-degree of any vertex equals its out-degree is Eulerian; this baseline result is used as the basis of existence proofs for universal cycles (also known as deBruijn cycles or $U$-cycles) of several…
Dyck paths (also balanced brackets and Dyck words) are among the most heavily studied Catalan families. This paper is a continuation of [2, 3]. In the paper we enumerate the terms of the OEIS A036991, Dyck numbers, and construct a…
We introduce the notion of unavoidable (complete) sets of word patterns, which is a refinement for that of words, and study certain numerical characteristics for unavoidable sets of patterns. In some cases we employ the graph of pattern…
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…
A subfamily of Dyck words called tight Dyck words is seen to correspond, via a "castling" procedure, to the vertex set of an ordered tree $T$. From $T$, a "blowing" operation recreates the whole family ol Dyck words. The vertices of $T$ can…
A connected digraph in which the in-degree of any vertex equals its out-degree is Eulerian, this baseline result is used as the basis of existence proofs for universal cycles (also known as generalized deBruijn cycles or U-cycles) of…
We define a set of binary matrices where any two of them can not be placed one on the other in a way such that the corresponding entries coincide. The rows of the matrices are obtained by means of Dyck words. The cardinality of the set of…
In this paper we give a bijective proof for a relation between uni- bi- and tricellular maps of certain topological genus. While this relation can formally be obtained using Matrix-theory as a result of the Schwinger-Dyson equation, we here…
We propose different ways of lifting the notion of Dyck language from words to 2-dimensional (2D) pictures, by means of new definitions of increasing comprehensiveness. Two of the proposals are based on alternative definitions of a Dyck…
Delta-matroid theory is often thought of as a generalization of topological graph theory. It is well-known that an orientable embedded graph is bipartite if and only if its Petrie dual is orientable. In this paper, we first introduce the…
There is a large class of problems in algebraic combinatorics which can be distilled into the same challenge: construct an explicit combinatorial bijection. Traditionally, researchers have solved challenges like these by visually inspecting…
A new algorithm to generate all Dyck words is presented, which is used in ranking and unranking Dyck words. We emphasize the importance of using Dyck words in encoding objects related to Catalan numbers. As a consequence of formulas used in…
Within this research, two combinatorial bijections using Young diagrams were studied. The first is a special case of a bijective correspondence between two classes of combinatorial objects. Its proof, based on Young diagrams, establishes…
A De Bruijn cycle is a cyclic sequence in which every word of length $n$ over an alphabet $\mathcal{A}$ appears exactly once. De Bruijn tori are a two-dimensional analogue. Motivated by recent progress on universal partial cycles and words,…
Recently, Sagan and Savage introduced the notion of Eulerian pairs. In this note, we find Eulerian pairs on Fibonacci words based on Foata's first transformation or Han's bijection and a map in the spirit of a bijection of…
Many "good" topologies for interconnection networks are based on line digraphs of regular digraphs. These digraphs support unitary matrices. We propose the property "being the digraph of a unitary matrix" as additional criterion for the…
In this note, we provide a bijection between a new collection of words on nonnegative integers of length n and Dyck paths of length 2n-2, thus proving that this collection belongs to the Catalan family. The surprising key step in this…
We extend Schaeffer's bijection between rooted quadrangulations and well-labeled trees to the general case of Eulerian planar maps with prescribed face valences, to obtain a bijection with a new class of labeled trees, which we call…
We propose a simple new combinatorial model to study spaces of acyclic Jacobi diagrams, in which they are identified with algebras of words modulo operations. This provides a starting point for a word-problem type combinatorial…
There is a natural bijection between Dyck paths and basis diagrams of the Temperley-Lieb algebra defined via tiling. Overhang paths are certain generalisations of Dyck paths allowing more general steps but restricted to a rectangle in the…