Counting strings in Dyck paths

Narayana numbers and Schur–Szegő composition

“…A polynomial P is called hyperbolic if all its roots are real. Narayana numbers recently appeared in a number of different combinatorial situations, see [5,11,15]. Remark 1.…”

Equivalence classes of permutations modulo descents and left-to-right maxima

“…For instance, the Dyck path U U U U DDDD would produce π = 4123 / ∈ Irr 4 , then applying the described process, the permutation σ = 4321 ∈ Irr 4 is obtained. Finally, the result is obtained since the set of Dyck paths with no occurrence of U DU D is enumerated by the sequence A078481 in [18] (see [17]). …”

Pattern avoiding ballot paths and finite operator calculus

“…In their paper Counting Strings in Dyck Paths [6], A. Sapounakis, I. Tasoulas, and P. Tsikouras find generating functions for all patterns of length 4 occurring k times in Dyck paths, i.e., in ballot paths returning to the diagonal. Their case k = 0 is our pattern avoiding case.…”

“…Dyck paths containing k strings of length 3 were discussed by E. Deutsch in [1]. One of the most recent papers on patterns of length 4 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 [6]. The authors find generating functions for all 16 patterns in Dyck path returning to the x-axis.…”

“…(6,8) shows the first instance when the occurrence of the pattern is subtracted. The double path (⇒) to (9, 11) shows the first instance when the occurrence of the op ′ path is added.…”