Related papers: Enumeration of \L{}ukasiewicz paths modulo some pa…
The classical Chung-Feller theorem [2] tells us that the number of Dyck paths of length $n$ with flaws $m$ is the $n$-th Catalan number and independent on $m$. L. Shapiro [7] found the Chung-Feller properties for the Motzkin paths. In this…
We consider a variation of Dyck paths, where additionally to steps $(1,1)$ and $(1,-1)$ down-steps $(1,-j)$, for $j\ge2$ are allowed. We give credits to Emeric Deutsch for that. The enumeration of such objects living in a strip is…
In this paper we compare the Katz similarity index to two other node similarity metrics, the standard distance and the resistance distance, for both path and cycle graphs. We consider how Katz similarity varies as the parameter $\alpha$ and…
Let $\mathcal{L}_n$ denote the set of all paths from $[0,0]$ to $[n, n]$ which consist of either unit north steps $N$ or unit east steps $E$ or, equivalently, the set of all words $L \in \{E,N\}^*$ with $n$ $E$'s and $n$ $N$'s. Given $L \in…
Two mesh patterns are coincident if they are avoided by the same set of permutations, and are Wilf-equivalent if they have the same number of avoiders of each length. We provide sufficient conditions for coincidence of mesh patterns, when…
An exponential automorphism of $\mathbf{C}$ is a function $\alpha: \mathbf{C} \rightarrow \mathbf{C}$ such that $\alpha(z_1 + z_2) = \alpha(z_1) + \alpha(z_2)$ and $\alpha\left( e^z \right) = e^{\alpha(z)}$ for all $z, z_1, z_2 \in…
For $\ell \geq 1$ and $k \geq 2$, we consider certain admissible sequences of $k-1$ lattice paths in a colored $\ell \times \ell$ square. We show that the number of such admissible sequences of lattice paths is given by the sum of squares…
In this paper, we study symmetric lattice paths. Let $d_{n}$, $m_{n}$, and $s_{n}$ denote the number of symmetric Dyck paths, symmetric Motzkin paths, and symmetric Schr\"oder paths of length $2n$, respectively. By using Riordan group…
When two patterns occur equally often in a set of permutations, we say that these patterns are equipopular. Using both structural and analytic tools, we classify the equipopular patterns in the set of separable permutations. In particular,…
Ternary paths consist of an up-step of one unit, a down-step of two units, never go below the $x$-axis, and return to the $x$-axis. This paper addresses the enumeration of partial ternary paths, ending at a given level $i$, reading the path…
BCK-sequences and n-commutative BCK-algebras were introduced by T. Traczyk, together with two related problems. The first one, whether BCK-sequences are always prolongable. The second one, if the class of all n-commutative BCK-algebras is…
Let $G$ be a $k$-connected graph with $k\geq 2$. In this paper we first prove that: For two distinct vertices $x$ and $z$ in $G$, it contains a path passing through its any $k-2$ {specified} vertices with length at least the average degree…
The set of discrete lattice paths from (0, 0) to (n, n) with North and East steps (i.e. words w $\in$ { x, y } * such that |w| x = |w| y = n) has a canonical monoid structure inherited from the bijection with the set of join-continuous maps…
This is a review of several results related to distribution of powers and combination of powers modulo 1. We include a proof that given a sequence of real numbers $\theta_n$, it is possible to get an $\alpha$ (given $\lambda \ne 0$), or a…
Inhomogeneous lattice paths are introduced as ordered sequences of rectangular Young tableaux thereby generalizing recent work on the Kostka polynomials by Nakayashiki and Yamada and by Lascoux, Leclerc and Thibon. Motivated by these works…
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 derive a path counting formula for two-dimensional lattice path model with filter restrictions in the presence of long steps, source and target points of which are situated near the filters. This solves a problem of finding an explicit…
For any positive integer $n$ we describe the Leavitt path algebra of the Cayley graph $C_n$ corresponding to the cyclic group $\Z/n\Z$. Using a Kirchberg-Phillips-type realization result, we show that there are exactly four isomorphism…
We enumerate the edges in the Hasse diagram of several lattices arising in the combinatorial context of lattice paths. Specifically, we will consider the case of Dyck, Grand Dyck, Motzkin, Grand Motzkin, Schr\"oder and Grand Schr\"oder…
Contrary to previous approaches bringing together algebraic geometry and signatures of paths, we introduce a Zariski topology on the space of paths itself, and study path varieties consisting of all paths whose iterated-integrals signature…