Equivalence classes of ballot paths modulo strings of length 2 and 3

Abstract

a b s t r a c tTwo paths are equivalent modulo a given string τ , whenever they have the same length and the positions of the occurrences of τ are the same in both paths. This equivalence relation was introduced for Dyck paths in , where the number of equivalence classes was evaluated for any string of length 2.In this paper, we evaluate the number of equivalence classes in the set of ballot paths for any string of length 2 and 3, as well as in the set of Dyck paths for any string of length 3.

Dyck paths with catastrophes modulo the positions of a given pattern

“…On the other hand, in [2][3][4][5][6]10] the authors investigate equivalence relations on the sets of Dyck paths, Motzkin paths, skew Dyck paths, Lukasiewicz paths, and Ballot paths where two paths of the same length are equivalent whenever they coincide on all occurrences of a given pattern. The main goal of this study consists in extending these studies for Dyck paths with catastrophes by considering the analogous equivalence relation on E.…”

Bijections from Dyck and Motzkin meanders with catastrophes to pattern avoiding Dyck paths

“…Recently in [1], Banderier and Wallner provide many results about the enumeration and limit laws of these objects. Using algrebraic methods they prove that the set M n of length n Dyck meanders with catastrophes has the same cardinality as the set of equivalence classes of semilength n + 1 Dyck paths modulo the positions of the pattern DU U , which in turn (see [10]) is in one-to-one correspondence with the set A n of semilength n Dyck paths avoiding occurrences at height h > 0 of the patterns U U U and DU D. They also provide a constructive bijection between E n and the set of length n Motzkin paths having their flat steps F at height one.…”

Enumeration of Łukasiewicz paths modulo some patterns

“…It consists in determining the cardinality of the quotient set generated by an equivalence relation based on the positions of a given pattern: two paths belong to the same equivalence class whenever the positions of occurrences of a given pattern are identical on these paths. Enumerating results are provided for the quotient sets of Dyck, Motzkin and Ballot paths for patterns of length at most three (see respectively [2], [3] and [13]). The purpose of this present paper is to extend these studies for Lukasiewicz paths that naturally generalizes Dyck and Motzkin paths.…”

“…Following the recent studies [2,3,13], we define an equivalence relation on the set L for a given pattern α: two Lukasiewicz paths of the same length are α-equivalent whenever the occurrences of the pattern α appear at the same positions in the two paths. For instance, UF F F FDUDUDF F F F UDF F is F D-equivalent to the path C in Figure 1 since the only one occurrence of the pattern F D (in boldface) appear at the same position in the two paths.…”