Abstract: Consider lattice paths in the plane allowing the steps (1,1), (1,-1), and (w,0), for some nonnegative integer w. For n > 1, let E(n,0) denote the set of paths from (0,0) to (n,0) running strictly above the x-axis except initially and finally. Generating functions are given for sums of moments of the ordinates of the lattice points on the paths in E(n,0). In particular, recurrencess ar...

We consider plane trees whose vertices are given labels from the set {1, 2, . . . , k} in such a way that the sum of the labels along any edge is at most k + 1; it turns out that the enumeration of these trees leads to a generalization of the Catalan numbers. We also provide bijections between this class of trees and (k + 1)-ary trees as well as generalized Dyck paths whose step sizes are k (up...

We give an involution on the set of lattice paths from (0, 0) to (a, b) with steps N = (0, 1) and E = (1, 0) that lie between two boundaries T and B, which proves that the statistics ‘number of E steps shared with T ’ and ‘number of E steps shared with B’ have a symmetric joint distribution on this set. This generalizes a result of Deutsch for the case of Dyck paths.

We give a bijective proof of a conjecture of Regev and Vershik [7] on the equality of two multisets of hook numbers of certain skew-Young diagrams. The bijection proves a result that is stronger and more symmetric than the original conjecture, by means of a construction involving Dyck paths, a particular type of lattice path.

The purpose of this work is to extend the theory of finite operator calculus to the multivariate setting, and apply it to the enumeration of certain lattice paths. The lattice paths we consider are ballot paths. A ballot path is a path that stays weakly above the diagonal y = x, starts at the origin, and takes steps from the set {↑,→} = {u, r}. Given a string p from the set {u, r}∗, we want to ...

P. Chase and F. Ruskey each published a Gray code for length n binary strings with m occurrences of 1, coding m-combinations of n objects, which is two-close—that is, in passing from one binary string to its successor a single 1 exchanges positions with a 0 which is either adjacent to the 1 or separated from it by a single 0. If we impose the restriction that any suffix of a string contains at ...

In this paper we consider the problem of deciding membership in Dyck languages, a fundamental family of context-free languages, comprised of well-balanced strings of parentheses. In this problem we are given a string of length n in the alphabet of parentheses of m types and must decide if it is well-balanced. We consider this problem in the property testing setting, where one would like to make...

a graph is textit{symmetric}, if its automorphism group is transitive on the set of its arcs. in this paper, we  classifyall the connected cubic symmetric  graphs of order $36p$  and $36p^{2}$, for each prime $p$, of which the proof depends on the classification of finite simple groups.

We characterize the sets of centrosymmetric permutations, namely, permutations σ ∈ Sn such that σ(i)+σ(n+1−i) = n+1, that avoid any given family of patterns of length 3. We exhibit bijections between some sets of restricted centrosymmetric permutations and sets of classical combinatorial objects, such as Dyck prefixes and subsets of [n] containing no consecutive integers.

Skew Dyck paths without up–down–left are enumerated. In a second step, the number of contiguous subwords ‘up–down–left’ counted. This explains and extends results that were posted in Encyclopedia Integer Sequences.