Related papers: Anchored Dyck Paths
We introduce rational Dyck tilings, or $(a,b)$-Dyck tilings, and study them by the decomposition into $(1,1)$-Dyck tilings. This decomposition allows us to make use of combinatorial models for $(1,1)$-Dyck tilings such as the Hermite…
We introduce the notion of doubly rooted plane trees and give a decomposition of these trees, called the butterfly decomposition which turns out to have many applications. From the butterfly decomposition we obtain a one-to-one…
We describe arithmetic algorithms on a canonical number representation based on the Catalan family of combinatorial objects specified as a Haskell type class. Our algorithms work on a {\em generic} representation that we illustrate on…
In this paper we establish plenty of number theoretic and combinatoric identities involving generalized Bernoulli and Stirling numbers of both kinds. These formulas are deduced from Pascal type matrix representations of Bernoulli and…
We present combinatorial bijections and identities between certain skew Young tableaux, Dyck paths, triangulations, and dissections.
The Catalan numbers count many classes of combinatorial objects. The most emblematic such objects are probably the Dyck walks and the binary trees, and, whenever another class of combinatorial objects is counted by the Catalan numbers, it…
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 substantial generalization of the equinumeracy of grand Dyck paths and Dyck-path prefixes, constrained within a band. The number of constrained paths starting at level $i$ and ending in a window of size $2j+2$ is equal to the…
Dyck tilings are certain tilings in the region surrounded by two Dyck paths. We study bijections and combinatorial objects bijective to Dyck tilings, which include Dyck tiling strip (DTS) and Dyck tiling ribbon (DTR) bijections, increasing…
We establish a tantalizing symmetry of certain numbers refining the Narayana numbers. In terms of Dyck paths, this symmetry is interpreted in the following way: if $w_{n,k,m}$ is the number of Dyck paths of semilength $n$ with $k$…
The classical Chung-Feller theorem [2] tells us that the number of Dyck paths of length $n$ with flaws $m$ is the $n$-th Catalan number and independent on $m$. L. Shapiro [7] found the Chung-Feller properties for the Motzkin paths. In this…
We study a class of observables in four-dimensional superconformal Yang--Mills theories which, in the planar limit at finite 't Hooft coupling, can be expressed as determinants of semi-infinite matrices built from Bessel functions. This…
The higher $q,t$-Catalan polynomial $C^{(m)}_n(q,t)$ can be defined combinatorially as a weighted sum of lattice paths contained in certain triangles, or algebraically as a complicated sum of rational functions indexed by partitions of $n$.…
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,…
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…
For each pair of coprime integers $a$ and $b$ we have a rational $q$-Catalan number $\operatorname{Cat}(a,b)_q=\binom{a+b}{a}_q/[a+b]_q$. It is known that this is a polynomial in $q$ with nonnegative integer coefficients, but the nature of…
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 study the Fuss--Catalan algebras, which are generalizations of the Temperley--Lieb algebra and act on generalized Dyck paths, through non-crossing partitions. First, the Temperley--Lieb algebra is defined on non-crossing partitions, and…
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…
We present a new alternating convolution formula for the super Catalan numbers which arises as a generalization of two known binomial identities. We prove a generalization of this formula by using auxiliary sums, recurrence relations, and…