A bijection between the sets of (a,b,b2)(a,b,b^2)-Generalized Motzkin paths avoiding uvv\mathbf{uvv}-patterns and uvu\mathbf{uvu}-patterns

A generalized Motzkin path, called G-Motzkin path for short, of length $n$ is a lattice path from $(0, 0)$ to $(n, 0)$ in the first quadrant of the XOY-plane that consists of up steps $\mathbf{u}=(1, 1)$, down steps $\mathbf{d}=(1, -1)$, horizontal steps $\mathbf{h}=(1, 0)$ and vertical steps $\mathbf{v}=(0, -1)$. An $(a,b,c)$-G-Motzkin path is a weighted G-Motzkin path such that the $\mathbf{u}$-steps, $\mathbf{h}$-steps, $\mathbf{v}$-steps and $\mathbf{d}$-steps are weighted respectively by $1, a, b$ and $c$. Let $\tau$ be a word on $\{\mathbf{u}, \mathbf{d}, \mathbf{v}, \mathbf{d}\}$, denoted by $\mathcal{G}_n^{\tau}(a,b,c)$ the set of $\tau$-avoiding $(a,b,c)$-G-Motzkin paths of length $n$ for a pattern $\tau$. In this paper, we consider the $\mathbf{uvv}$-avoiding $(a,b,c)$-G-Motzkin paths and provide a direct bijection $\sigma$ between $\mathcal{G}_n^{\mathbf{uvv}}(a,b,b^2)$ and $\mathcal{G}_n^{\mathbf{uvu}}(a,b,b^2)$. Finally, the set of fixed points of $\sigma$ is also described and counted.Comment: 11pages,2 figures. arXiv admin note: substantial text overlap with arXiv:2201.0923

On Directed Lattice Paths With Additional Vertical Steps

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. It is shown that for any family of primary paths there are equinumerous families of proper weighted lattice paths that consist of only non-vertical steps. The relation between primary and free paths is established and some combinatorial and statistical properties are obtained. It is shown that the expected number of vertical steps in a primary path running from $(0,0)$ to $(n,-1)$ is equal to the number of free paths running from $(0,0)$ to $(n,0)$. Enumerative results with generating functions are given. Finally, a few examples of families of paths with vertical steps are presented and related to {\L}ukasiewicz, Motzkin, Dyck and Delannoy paths.Comment: 23 pages, 6 figure

Asymmetric extension of Pascal-Dellanoy triangles

We give a generalization of the Pascal triangle called the quasi s-Pascal triangle where the sum of the elements crossing the diagonal rays produce the s-bonacci sequence. For this, consider a lattice path in the plane whose step set is {L = (1, 0), L1 = (1, 1), L2 = (2, 1), . . . , Ls = (s, 1)}; an explicit formula is given. Thereby linking the elements of the quasi s-Pascal triangle with the bisnomial coefficients. We establish the recurrence relation for the sum of elements lying over any finite ray of the quasi s-Pascal triangle. The generating function of the cited sums is produced. We also give identities among which one equivalent to the de Moivre sum and establish a q-analogue of the coefficient of the quasi s-Pascal triangle.Comment: 17 pages, 3 tables, 2 illustrations, 1 figur