Related papers: Down-step statistics in generalized Dyck paths
It is well known that the set of $m$-Dyck paths with a fixed height and a fixed amount of valleys is counted by the Fu{\ss}-Narayana numbers. In this article, we consider the set of $m$-Dyck paths that start with at least $t$ north steps.…
We address the problem of enumerating paths in square lattices, where allowed steps include (1,0) and (0,1) everywhere, and (1,1) above the diagonal y=x. We consider two such lattices differing in whether the (1,1) steps are allowed along…
We introduce a subfamily of skew Dyck paths called box paths and show that they are in bijection with pairs of ternary trees, confirming an observation stated previously on the On-Line Encyclopedia of Integer Sequences. More generally, we…
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…
We answer a question of Simental by providing a combinatorial interpretation of a formula which generalizes rational Catalan numbers and which appears in the study of Springer fibers. We provide an interpretation in terms of binary…
A variation of Dyck paths allows for down-steps of arbitrary length, not just one. Credits for this invention are given to Emeric Deutsch. Surprisingly, the enumeration of them is somewhat akin to the analysis of Motzkin-paths; the last…
A 3-dimensional Catalan word is a word on three letters so that the subword on any two letters is a Dyck path. For a given Dyck path $D$, a recently defined statistic counts the number of Catalan words with the property that any subword on…
We prove new bijections between different variants of Dyck paths and integer compositions, which give combinatorial explanations of their simple counting formula $4^{n-1}$. These give relations between different statistics, such as the…
We introduce the notion of pattern in the context of lattice paths, and investigate it in the specific case of Dyck paths. Similarly to the case of permutations, the pattern-containment relation defines a poset structure on the set of all…
Bargraphs are a special class of convex polyominoes. They can be identified with lattice paths with unit steps north, east, and south that start at the origin, end on the $x$-axis, and stay strictly above the $x$-axis everywhere except at…
Pappe, Paul, and Schilling introduced two combinatorial statistics, depth and ddinv, associated with classical Dyck paths, and proved that the distributions of (area, depth) and (dinv, ddinv) are $q,t$-symmetric by constructing an…
In a paper by Sapounakis, Tasoulas, and Tsikouras \cite{stt}, the authors count the number of occurrences of patterns of length four in Dyck paths. In this paper we specify in one direction and generalize in another. We only count ballot…
We consider the distribution of ascents, descents, peaks, valleys, double ascents, and double descents over permutations avoiding a set of patterns. Many of these statistics have already been studied over sets of permutations avoiding a…
Using the bijection between partitions and vacillating tableaux, we establish a correspondence between pairs of noncrossing free Dyck paths of length $2n$ and noncrossing partitions of $[2n+1]$ with $n+1$ blocks. In terms of the number of…
We show that the distribution of the number of peaks at height $i$ modulo $k$ in $k$-Dyck paths of a given length is independent of $i\in[0,k-1]$ and is the reversal of the distribution of the total number of peaks. Moreover, these…
Dispersed Dyck paths are Dyck paths, with possible flat steps on level 0. We revisit and augment questions about them from the Encyclopedia of Integer Sequences, in a systematic way that uses generating functions and the kernel method.
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…
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…
Recently, Kenyon and Wilson introduced Dyck tilings, which are certain tilings of the region between two Dyck paths. The enumeration of Dyck tilings is related with hook formulas for forests and the combinatorics of Hermite polynomials. The…
Skew Dyck paths without up-down-left are enumerated. In a second step, the number of contiguous subwords 'up-down-left' are counted. This explains and extends results that were posted in the Encyclopedia of Integer Sequences.