Euler Coefficients and Restricted Dyck Paths

One of the most recent papers on patterns occurring k times in Dyck paths was written by A. Sapounakis, I. Tasoulas, P. Tsikouras, Counting strings in Dyck paths, 2007, to appear in Discrete Mathematics [5]. The authors find generating functions for all 16 patterns generated by combinations of four up (ր) and down (ց) steps. A Dyck path starts at (0, 0), takes only up and down steps, and ends at (2n, 0), staying weakly above the x-axis. Returning to the x-axis at the end of the path has the advantage that every path containing the pattern uduu, say, k times, will contain the reversed pattern ddud also k times when read backwards. This reduces significantly the number of patterns under consideration. Dyck paths containing k strings of length 3 were discussed by E. Deutsch in [1]. In this paper we consider only the patterns u and d, for all integers r > 2, and we will investigate only the case k = 0, which means pattern avoidance. It has been shown in [5] that the generating function f(t) for avoiding u(or d) satisfies the equation f (t) = 1+ ∑r−1 i=1 t f (t) i = 1−t−t rf(t)r 1−2t . However, we will allow the Dyck paths to end at (n,m), m ≥ 0, which removes the above mentioned symmetry, as shown in the following two tables.