Related papers: Bijections between directed animals, multisets and…
Let $r(n,k)$ (resp. $s(n,k)$) be the number of Schr\"oder paths (resp. little Schr\"oder paths) of length $2n$ with $k$ hills, and set $r(0,0)=s(0,0)=1$. We bijectively establish the following recurrence relations: \begin{align*}…
A Dyck path is a lattice path in the plane integer lattice $\mathbb{Z}\times\mathbb{Z}$ consisting of steps (1,1) and (1,-1), which never passes below the x-axis. A peak at height k on a Dyck path is a point on the path with coordinate y=k…
In this note we introduce several instructive examples of bijections found between several different combinatorially defined sequences of sets. Each sequence has cardinalities given by the Catalan numbers. Our results answer some questions…
Arnol'd proved in 1992 that Springer numbers enumerate the Snakes, which are type $B$ analogs of alternating permutations. Chen, Fan and Jia in 2011 introduced the labeled ballot paths and established a ``hard'' bijection with snakes.…
We give a general construction of triangulations starting from a walk in the quarter plane with small steps, which is a discrete version of the mating of trees. We use a special instance of this construction to give a bijection between maps…
We study four bijections, which are promotion, evacuation, rowmotion, and rowvacuation, on generalized Dyck paths in rational Catalan combinatorics. We define the maps on generalized Dyck paths, which have their origins in maps on Dyck…
Set partitions avoiding $k$-crossing and $k$-nesting have been extensively studied from the aspects of both combinatorics and mathematical biology. By using the generating tree technique, the obstinate kernel method and Zeilberger's…
In this paper we build a bijection between the meet-irreducible elements of the lattice of the compositions of $n$ with parts in $[1,p]$ equipped with the dominance order, and the edges of the $(n,p)$-Tur\'an graph. Using this bijection, we…
We define $(k,\ell)$-restricted Lukasiewicz paths, $k\le\ell\in\mathbb{N}_0$, and use these paths as models of polymer adsorption. We write down a polynomial expression satisfied by the generating function for arbitrary values of…
Tandem duplication is an evolutionary process whereby a segment of DNA is replicated and proximally inserted. The different configurations that can arise from this process give rise to some interesting combinatorial questions. Firstly, we…
We provide a bijection between the set of factorizations, that is, ordered (n-1)-tuples of transpositions in ${\mathcal S}_{n}$ whose product is (12...n), and labelled trees on $n$ vertices. We prove a refinement of a theorem of D\'{e}nes…
A bijection is presented between (1): partitions with conditions $f_j+f_{j+1}\leq k-1$ and $ f_1\leq i-1$, where $f_j$ is the frequency of the part $j$ in the partition, and (2): sets of $k-1$ ordered partitions $(n^{(1)}, n^{(2)}, ...,…
We present a simple a bijection between permutations of $\{1,..., n\}$ with $k$ descents and permutation tableaux of length $n$ with $k$ columns.
We generalize the concept of ascending and descending runs from permutations to rooted labelled trees and mappings, i.e., functions from the set $\{1, \dots, n\}$ into itself. A combinatorial decomposition of the corresponding functional…
Randomly sampling an acyclic orientation on the complete bipartite graph $K_{n,k}$ with parts of size $n$ and $k$, we investigate the length of the longest path. We provide a probability generating function for the distribution of the…
Recently, Mansour and Shattuck related the total number of humps in all of the $(k, a)$-paths of order $n$ to the number of super $(k, a)$-paths, which generalized previous results concerning the cases when $k = 1$ and $a = 1$ or $a =…
We investigate pattern avoidance in permutations satisfying some additional restrictions. These are naturally considered in terms of avoiding patterns in linear extensions of certain forest-like partially ordered sets, which we call binary…
A generalized Motzkin path, called G-Motzkin path for short, of length $n$ is a lattice path from $(0, 0)$ to $(n, 0)$ in the first quadrant of the XOY-plane that consists of up steps $\mathbf{u}=(1, 1)$, down steps $\mathbf{d}=(1, -1)$,…
We investigate paths in the hexagonal circle packing and enumerate them with respect to width, height, number of steps, area, and kissing number. Functional equations and the kernel method yield closed bivariate generating functions…
We introduce weighted succession rules and parametric production matrices - simple extensions of the standard ECO method succession rules and production matrices. The purpose is to enumerate combinatorial objects with respect to several…