Related papers: Skew Dyck paths without up--down--left
The enumeration of k-Dyck paths ending at level j after m up-steps, where the last step is an up-step, is given as a sum, improving on a previous formula given by Deng and Mansour.
We consider the class of directed graphs with $N\geq 1$ edges and without loops shorter than $k\geq1$. Using the concept of a labelled graph, we determine graphs from this class that maximize the number of all paths of length $k$. Then we…
We give a bijective proof of a conjecture of Regev and Vershik on the equality of two multisets of hook numbers of certain skew-Young diagrams. The bijection proves a result that is stronger and more symmetric than the original conjecture,…
We consider the problem of counting subset of Dyck paths contained in a Ferrers diagram. This enumeration concerns to find the number of the elements in a branch of the Kr\'ew\'eras tree. Using the Ferrers diagrams associated with Dyck…
The simple connected graphs may be classified by their cycle composition (number and lengths of cycles). This work derives the counting series of the simple connected graphs that have cycles of unrestricted number and length, but no…
Consider a directed multigraph $D$ that is balanced (i.e., at each vertex, the indegree equals the outdegree). Let $A$ be its set of arcs. Fix an integer $k$. Let $s$ be a vertex of $D$. We show that the number of $k$-element subsets $B$ of…
Dyck paths having height at most $h$ and without valleys at height $h-1$ are combinatorially interpreted by means of 312-avoding permutations with some restrictions on their \emph{left-to-right maxima}. The results are obtained by analyzing…
An improved upper bound is obtained for the density of sequences of positive integers that contain no k-term geometric progression.
The downward closure of a language $L$ of words is the set of all (not necessarily contiguous) subwords of members of $L$. It is well known that the downward closure of any language is regular. Although the downward closure seems to be a…
The family of graphs of reduced words of a certain subcollection of permutations in the union $\cup_{n\geq 4}\frak{S}_{n}$ of symmetic groups is investigated. The subcollection is characterised by the hook cycle type $(n-2,1,1)$ with…
We study subshift that arise by excluding words of length two from Dyck shifts. The words that are to be excluded are taken from a finite set that is not literal-uniform.
Let $0<k\in\mathbb{Z}$. We zipper-merge integer compositions with sums $k$ and $k+1$, equal number of parts and initial entries equal at least to 1 and 2, respectively. This yields bitstrings with two initial zeros, $k-1$ remaining zeros…
A class of skew derivations on complex Noetherian generalized down-up algebras $L=L(f,r,s,\gamma)$ is constructed.
The lattice path model suggested by E. Deutsch is derived from ordinary Dyck paths, but with additional down-steps of size -3,-5,-7,... . For such paths, we find the generating functions of them, according to length, ending at level $i$,…
In this article, we provide a bijection between the set of inversion sequences avoiding the pattern 102 and the set of 2-Schr\"{o}der paths having neither peaks nor valleys and ending with a diagonal step. To achieve this, we introduce two…
A dispersed Dyck path (DDP) of length n is a lattice path on $N\times N$ from (0,0) to (n,0) in which the following steps are allowed: "up" (x, y) $\to$ (x+1, y+1); "down" (x, y) $\to$ (x+1, y-1); and "right" (x,0) $\to$ (x+1,0). An ascent…
A descent of a labeled acyclic digraph is a directed edge $x\to y$ with $x>y$. In this paper, we find a recurrence for the number of labeled acyclic digraphs with a given number of descents.
We show that our algorithm for inverting the sweep map on (2n, n)-Dyck paths works for any (kn, n)-Dyck path, where k is an arbitrary positive integer.
Motzkin excursions and meanders are revisited. This is considered in the context of forbidden patterns. Previous work by Asinowski, Banderier, Gittenberger, and Roitner is continued. Motzkin paths of bounded height are considered, leading…
In this paper, we show that both 12312-avoiding partitions and 12321-avoiding partitions of the set $[n+1]$ are in one-to-one correspondence with Schr\"oder paths of semilength $n$ without peaks at even level. As a consequence, the refined…