Related papers: Raney numbers, threshold sequences and Motzkin-lik…
The authors propose a new variation of random walks called ladder chains $L(r,s,p)$. We extend concepts such as ruin probability, hitting time, transience and recurrence of random walks to ladder chain. Take $L(2,2,p)$ for instance, we find…
Linear recursions of degree $k$ are determined by evaluating the sequence of Generalized Fibonacci Polynomials, $\{F_{k,n}(t_1,...,t_k)\}$ (isobaric reflects of the complete symmetric polynomials) at the integer vectors $(t_1,...,t_k)$. If…
We provide generating functions, formulas, and asymptotic expressions for the number of Catalan words based on the number of runs of ascents (descents), runs of weak ascents (descents), $\ell$-valleys, valleys, symmetric valleys,…
During the study of dual sequences, Sun introduced the polynomials \[ D_n(x,y)=\sum_{k=0}^{n}{n\choose k}{x\choose k}y^k\text{ and } S_n(x,y)=\sum_{k=0}^{n}\binom{n}{k}\binom{x}{k}\binom{-1-x}{k} y^k. \] Many related congruences have been…
For a wide class of integer linear recurrence sequences $\left(u(n)\right)_{n=1}^\infty$, we give an upper bound on the number of $s$-tuples $\left(n_1, \ldots, n_s\right) \in \left(\mathbb Z\cap [M+1,M+ N]\right)^s$ such that the…
Let $a,b$ be fixed positive coprime integers. For a positive integer $g$, write $W_k(g)$ for the set of lattice paths from the startpoint $(0,0)$ to the endpoint $(ga,gb)$ with steps restricted to $\{(1,0), (0,1)\}$, having exactly $k$…
Odlyzko and Stanley introduced a greedy algorithm for constructing infinite sequences with no 3-term arithmetic progressions when beginning with a finite set with no 3-term arithmetic progressions. The sequences constructed from this…
In this paper, we perform an arithmetization of well-formed parenthesis strings with zeros (Motzkin words) and of corresponding Motzkin paths. The transformations used are reminiscent of G\"odel numbering for mathematical objects of some…
In this paper we study the number of humps (peaks) in Dyck, Motzkin and Schr\"{o}der paths. Recently A. Regev noticed that the number of peaks in all Dyck paths of order $n$ is one half of the number of super Dyck paths of order $n$. He…
Assume that we embed the path $P_n$ as a subgraph of a $2$-dimensional grid, namely, $P_k \times P_l$. Given such an embedding, we consider the ordered set of subpaths $L_1, L_2, \ldots , L_m$ which are maximal straight segments in the…
The Gessel number $P(n,r)$ is the number of the paths in plane with $(1, 0)$ and $(0,1)$ steps from $(0,0)$ to $(n+r, n+r-1)$ that never touch any of the points from the set $\{(x,x)\in \mathbb{Z}^2: x\geq r\}$. We show that there is a…
Three $q$-versions of Lommel polynomials are studied. Included are explicit representations, recurrences, continued fractions, and connections to associated Askey--Wilson polynomials. Combinatorial results are emphasized, including a…
We present a method to obtain congruences modulo powers of 2 for sequences given by recurrences of finite depth with polynomial coefficients. We apply this method to Catalan numbers, Fu\ss-Catalan numbers, and to subgroup counting functions…
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.…
Given an alphabet $S$, we consider the size of the subsets of the full sequence space $S^{\rm {\bf Z}}$ determined by the additional restriction that $x_i\not=x_{i+f(n)},\ i\in {\rm {\bf Z}},\ n\in {\rm {\bf N}}.$ Here $f$ is a positive,…
For two graphs $G$ and $H$, let $r(G,H)$ and $r_*(G,H)$ denote the Ramsey number and star-critical Ramsey number of $G$ versus $H$, respectively. In 1996, Li and Rousseau proved that $r(K_{m},F_{t,n})=tn(m-1)+1$ for $m\geq 3$ and…
In a graph whose vertices are assigned integer ranks, a path is well-ranked if the endpoints have distinct ranks or some interior point has a higher rank than the endpoints. A ranking is an assignment of ranks such that all nontrivial paths…
We introduce an inductively defined sequence of directed graphs and prove that the number of edges added at step $k$ is equal to the $k$th Catalan number. Furthermore, we establish an isomorphism between the set of edges adjoined at step…
We say that a permutation $\pi$ is a Motzkin permutation if it avoids 132 and there do not exist $a<b$ such that $\pi_a<\pi_b<\pi_{b+1}$. We study the distribution of several statistics in Motzkin permutations, including the length of the…
We introduce the notion of infinitely log-monotonic sequences. By establishing a connection between completely monotonic functions and infinitely log-monotonic sequences, we show that the sequences of the Bernoulli numbers, the Catalan…