Related papers: Skew Dyck paths having no peaks at level 1
A \emph{Dyck path} is a lattice path in the first quadrant of the $xy$-plane that starts at the origin, ends on the $x$-axis, and consists of the same number of North-East steps $U$ and South-East steps $D$. A \emph{valley} is a subpath of…
In this paper we study a subfamily of a classic lattice path, the \emph{Dyck paths}, called \emph{restricted $d$-Dyck} paths, in short $d$-Dyck. A valley of a Dyck path $P$ is a local minimum of $P$; if the difference between the heights of…
\L{}ukasiewicz paths are lattice paths in $\Bbb{N}^2$ starting at the origin, ending on the $x$-axis, and consisting of steps in the set $\{(1,k), k\geq -1\}$. We give generating function and exact value for the number of $n$-length…
Dyck paths where peaks are only allowed on level 1 and on even-indexed levels, were introduced by Retakh and analysed by Zeilberger, with assistance from Ekhad. We add some combinatorial comments to the enumeration, which involves Motzkin…
A variation of Dyck paths allows for down-steps of arbitrary length, not just one. This is motivated by ideas due to Emeric Deutsch. We use the adding-a-new-slice technique and the kernel method to compute the number of maximal runs of…
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 show connection between Dyck paths with peaks of bounded height and random walks. The correspondence between a certain class of random walks and such Dyck paths allows us to develop a probabilistic perspective on Chebyshev polynomials.
A variation of Dyck paths allows for down-steps of arbitrary length, not just one. This is motivated by ideas published by Emeric Deutsch around the turn of the millenium. We are interested in the subclass of them where the sequence of the…
For $0\leq k\leq n-1$, we introduce a family of $k$-skeletal paths which are counted by the $n$-th Catalan number for each $k$, and specialize to Dyck paths when $k=n-1$. We similarly introduce $k$-skeletal parking functions which are…
The paper is devoted to the study of lattice paths that consist of vertical steps $(0,-1)$ and non-vertical steps $(1,k)$ for some $k\in \mathbb Z$. Two special families of primary and free lattice paths with vertical steps are considered.…
There was recent interest in Motzkin paths without peaks (peak: up-step followed immediately by down-step); additional results about this interesting family is worked out. The new results are the enumeration of such paths that live in a…
We consider Dyck paths having height at most two with some constraints on the number of consecutive valleys at height one which must be followed by a suitable number of valleys at height zero. We prove that they are enumerated by so-called…
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 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…
It is known that both the number of Dyck paths with $2n$ steps and $k$ peaks, and the number of Dyck paths with $2n$ steps and $k$ steps at odd height follow the Narayana distribution. In this paper we present a bijection which explicitly…
In this paper, we show that both 12312-avoiding partitions and 12321-avoiding partitions of the set $[n+1]$ are in one-to-one correspondence with Schr\"oder paths of semilength $n$ without peaks at even level. As a consequence, the refined…
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…
A rook path is a path on lattice points in the plane in which any proper horizontal step to the right or vertical step north is allowed. If, in addition, one allow bishop steps, that is, proper diagonal steps of slope 1, then one has queen…
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 paper concentrates on the set $\mathcal{V}_n$ of weighted Dyck paths of length $2n$ with special restrictions on the level of valleys. We first give its explicit formula of the counting generating function in terms of certain weight…