Partial Motzkin paths with air pockets of the first kind avoiding peaks, valleys or double rises (original) (raw)

Two kinds of partial Motzkin paths with air pockets

arXiv (Cornell University), 2022

Motzkin paths with air pockets (MAP) are defined as a generalization of Dyck paths with air pockets by adding some horizontal steps with certain conditions. In this paper, we introduce two generalizations. The first one consists of lattice paths in N 2 starting at the origin made of steps U = (1, 1), D k = (1, −k), k 1 and H = (1, 0), where two down steps cannot be consecutive, while the second one are lattice paths in N 2 starting at the origin, made of steps U , D k and H, where each step D k and H is necessarily followed by an up step, except for the last step of the path. We provide enumerative results for these paths according to the length, the type of the last step, and the height of its end-point. A similar study is made for these paths read from right to left. As a byproduct, we obtain new classes of paths counted by the Motzkin numbers. Finally, we express our results using Riordan arrays.

Enumeration of Dyck paths with air pockets

2022

We introduce and study the new combinatorial class of Dyck paths with air pockets. We exhibit a bijection with the peakless Motzkin paths which transports several pattern statistics and give bivariate generating functions for the distribution of patterns as peaks, returns and pyramids. Then, we deduce the popularities of these patterns and point out a link between the popularity of pyramids and a special kind of closed smooth self-overlapping curves, a subset of Fibonacci meanders. A similar study is conducted for the subclass of non-decreasing Dyck paths with air pockets.

Grand Dyck paths with air pockets

The Art of Discrete and Applied Mathematics

Grand Dyck paths with air pockets (GDAP) are a generalization of Dyck paths with air pockets by allowing them to go below the x-axis. We present enumerative results on GDAP (or their prefixes) subject to various restrictions such as maximal/minimal height, ordinate of the last point and particular first return decomposition. In some special cases we give bijections with other known combinatorial classes.

Restricted Dyck Paths on Valleys Sequence

2021

Abstract. In this paper we study a subfamily of a classic lattice path, the Dyck paths, called 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 two consecutive valleys (from left to right) is at least d, we say that P is a restricted d-Dyck path. The area of a Dyck path is the sum of the absolute values of y-components of all points in the path. We find the number of peaks and the area of all paths of a given length in the set of d-Dyck paths. We give a bivariate generating function to count the number of the d-Dyck paths with respect to the the semi-length and number of peaks. After that, we analyze in detail the case d = −1. Among other things, we give both, the generating function and a recursive relation for the total area.

Staircase tilings and lattice paths

We define a combinatorial structure, a tiling of the staircase in the R 2 plane, that will allow us, when restricted in different ways, to create direct bijections to Dyck paths of length 2n, Motzkin paths of lengths n and n−1, as well as Schröder paths and little Schröder paths of length n.

Motzkin Paths With a Restricted First Return Decomposition

Integers, 2019

Recently, the authors introduced new families of Dyck paths having a first decomposition constrained by the height or by the number of returns. In this work we extend the study to Motzkin paths and 2-colored Motzkin paths. For these new sets, we provide enumerative results by giving bivariate generating functions with respect to the length and another parameter, and we construct one-to-one correspondences with several restricted classes of ordered trees. We also deal with Schröder and Riordan paths. As a byproduct, we present a bijective proof of M(x)2 = 1 1−2xM( x 1−2x), where M(x) is the generating function of Motzkin numbers.

Dyck Paths with Peak- and Valley-Avoiding Sets

Studies in Applied Mathematics, 2008

In this paper, we focus on Dyck paths with peaks and valleys, avoiding an arbitrary set of heights. The generating functions of such types of Dyck paths can be represented by continued fractions. We also discuss a special case that requires all peak and valley heights to avoid congruence classes modulo k. We study the shift equivalence on sequences, which in turn induces an equivalence relation on avoiding sets.