Related papers: Bijections between Variants of Dyck Paths and Inte…
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…
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 give a bijection between partially directed paths in the symmetric wedge y= +/-x and matchings, which sends north steps to nestings. This gives a bijective proof of a result of Prellberg et al. that was first discovered through the…
This article deals with the enumeration of directed lattice walks on the integers with any finite set of steps, starting at a given altitude $j$ and ending at a given altitude $k$, with additional constraints such as, for example, to never…
For a given finite subset P of points of the lattice Z^2, a friendly path is a monotone (uphill or downhill) lattice path which splits points in half; points lying on the path itself are discarded. The purpose of this paper (and its sequel)…
For $0\leq k \leq n$, the number $C(n,k)$ represents the number of all lattice paths in the plane from the point $(0,0)$ to the point $(n,k)$, using steps $(1,0)$ and $(0,1)$, that never rise above the main diagonal $y=x$. The Fuss-Catalan…
We enumerate bijectively the family of involutive Baxter permutations according to various parameters; in particular we obtain an elementary proof that the number of involutive Baxter permutations of size $2n$ with no fixed points is…
A bijection is given between multi-edge trees and 3-coloured Motzkin paths.
We show that there is a bijection between real-linear automorphisms of the multicomplex numbers of order $n$ and signed permutations of length $2^{n-1}$. This allows us to deduce a number of results on the multicomplex numbers, including a…
We consider a variation of Dyck paths, where additionally to steps $(1,1)$ and $(1,-1)$ down-steps $(1,-j)$, for $j\ge2$ are allowed. We give credits to Emeric Deutsch for that. The enumeration of such objects living in a strip is…
The set of Dyck paths of length $2n$ inherits a lattice structure from a bijection with the set of noncrossing partitions with the usual partial order. In this paper, we study the joint distribution of two statistics for Dyck paths:…
We provide enumerating results for partial knight's paths of a given size. We prove algebraically that zigzag knight's paths of a given size ending on the $x$-axis are enumerated by the generalized Catalan numbers, and we give a…
We present a bijection between permutation matrices and descending plane partitions without special parts, which respects the quadruple of statistics considered by Behrend, Di Francesco and Zinn--Justin. This bijection involves the…
We give several bijections among restricted Motzkin paths, explaining why various parameters on these paths are equidistributed. For example, the number of doublerise-free Motzkin paths of length n is the same as the number of peak-free…
We enumerate the edges in the Hasse diagram of several lattices arising in the combinatorial context of lattice paths. Specifically, we will consider the case of Dyck, Grand Dyck, Motzkin, Grand Motzkin, Schr\"oder and Grand Schr\"oder…
We count the number of occurrences of restricted patterns of length 3 in permutations with respect to length and the number of cycles. The main tool is a bijection between permutations in standard cycle form and weighted Motzkin paths.
We investigate the definability (reducts) lattice of the order of integers and describe a sublattice generated by relations 'between', 'cycle', 'separation', 'neighbor', '1-codirection', 'order' and equality'. Some open questions are…
In this article we give a computational study of combinatorics of the discriminantal arrangements. The discriminantal arrangements are parametrized by two positive integers n and k such that n>k. The intersection lattice of the…
A lattice path inside the $m\times n$ table $T$ is a sequence $\nu_1,\ldots,\nu_k$ of cells such that $\nu_{j+1}-\nu_j\in\{(1,-1),(1,0),(1,1)\}$ for all $j=1,\ldots,k-1$. The number of lattice paths in $T$ from the first column to the…
It is known that for the 2n-step symmetric simple random walk on Z, two events have the same probability if and only if their sets of paths have the same cardinality. In this article, we construct two kinds of bijections between sets of…