EQUIVALENCE CLASSES OF SKEW DYCK PATHS MODULO SOME PATTERNS Jean-Luc Baril 1 LIB, Univ. Franche-Comté, France barjl@u-bourgogne.fr José Luis Ramírez Departamento de Matemáticas, Universidad Nacional de Colombia, Bogotá, Colombia jlramirezr@unal.edu.co Lina Maria Simbaqueba Departamento de Matemáticas, Universidad Nacional de Colombia, Bogotá, Colombia lmsimbaquebam@unal.edu.co Received: , Revised: , Accepted: , Published: Abstract For any pattern p ∈{U,L,UU,UD,DU,DD}, we enumerate the equivalence classes of skew Dyck paths, where two skew Dyck paths of the same semilength are p-equivalent whenever the positions of the occurrences of the pattern p are the same. In this paper we use generating functions, bijective arguments, and recurrence relations to obtain the main results. Keywords: Skew Dyck path, pattern, equivalence relation, generating function. 1. Introduction and notation A skew Dyck path is a lattice path in the first quadrant of the xy-plane that starts at the origin, ends on the x-axis, and made of up-steps U =(1, 1), down-steps D =(1, −1), and left steps L = (−1, −1) so that up and left steps do not overlap. Whenever we do not permit the step L, we retrieve the well known definition of Dyck paths (see [5]). We let SD denote the set of all skew Dyck paths, D the set of Dyck paths, and ∣P ∣ the length of the path P , i.e., the number of its steps, which is an even non-negative integer. Let λ be the skew Dyck path of length zero. For example, Figure 1 shows all skew Dyck paths of length 6, or equivalently of semilength 3. Figure 1: Skew Dyck paths of semilength 3. 1 Corresponding author