Related papers: Anchored Dyck Paths

Motivated by independent results of Bizley and Duchon, we study rational Dyck paths and their subset of factor-free elements. On the one hand, we give a bijection between rational Dyck paths and regular Dyck paths with ascents colored by…

We introduce a combinatorial enumeration problem that is solved using generalized Catalan numbers. We also study generalizations of the Cycle Lemma beyond the computation of the generalized Catalan numbers.

We study the two statistics, the inversion number and the major index, on Catalan combinatorial objects such as $r$-Dyck paths, $r$-Stirling permutations, non-crossing partitions, Dyck tilings, and symmetric Dyck paths. We show that they…

We announce a series of results on the combinatorial study of the q-Catalan triangle (C_{n,k}(q)), defined by C_{n,0}(q)=q^{n(n-1)/2} and C_{n,k}(q)=C_{n,k-1}(q)+q^{n-k-1}C_{n-1,k}(q). We establish combinatorial interpretations via a…

We introduce a notion of Dyck paths with coloured ascents. For several ways of colouring, we establish bijections between sets of such paths and other combinatorial structures, such as non-crossing trees, dissections of a convex polygon,…

The \emph{$q,t$-Catalan numbers} $C_n(q,t)$ are polynomials in $q$ and $t$ that reduce to the ordinary Catalan numbers when $q=t=1$. These polynomials have important connections to representation theory, algebraic geometry, and symmetric…

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…

We investigate certain nonassociative binary operations that satisfy a four-parameter generalization of the associative law. From this we obtain variations of the ubiquitous Catalan numbers and connections to many interesting combinatorial…

Given a Stirling permutation w, we introduce the mesa set of w as the natural generalization of the pinnacle set of a permutation. Our main results characterize admissible mesa sets and give closed enumerative formulas in terms of rational…

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…

Analysis of the dynamics of the Dyck words helped solve the problem of representing the Catalan number as a sum of squares of natural numbers. In this case, the Dyck triangle is considered in different coordinates. In the calculations, we…

We show that recent determinant evaluations involving Catalan numbers and generalisations thereof have most convenient explanations by combining the Lindstr\"om-Gessel-Viennot theorem on non-intersecting lattice paths with a simple…

Given a positive rational $q$, we consider Dyck paths having height at most two with some constraints on the number of consecutive peaks and consecutive valleys, depending on $q$. We introduce a general class of Dyck paths, called rational…

We investigate a natural Heyting algebra structure on the set of Dyck paths of the same length. We provide a geometrical description of the operations of pseudocomplement and relative pseudocomplement, as well as of regular elements. We…

Let $w$ be a word in alphabet $\{x,D\}$ with $m$ $x$'s and $n$ $D$'s. Interpreting "$x$" as multiplication by $x$, and "$D$" as differentiation with respect to $x$, the identity $wf(x) = x^{m-n}\sum_k S_w(k) x^k D^k f(x)$, valid for any…

Weighted Catalan numbers are a class of weighted sums over Dyck paths. Well-studied for their arithmetic properties and applications to enumerative combinatorics, these numbers were recently generalized to the setting of $k$-dimensional…

We use results on Dyck words and lattice paths to derive a formula for the exact number of binary words of a given length with a given minimal abelian border length, tightening a bound on that number from Christodoulakis et al. (Discrete…

In this paper, we present an involution on some kind of colored $k$-ary trees which provides a combinatorial proof of a combinatorial sum involving the generalized Catalan numbers $C_{k,\gamma}(n)=\frac{\gamma}{k n+\gamma}{k n+\gamma\choose…

We define new generalizations of (q,t)-Catalan numbers applying nabla operator on k-Schur functions indexed by column partitions. In some special cases, we give a combinatorial interpretation of these numbers using configurations of Dyck…

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…