Some statistics on Dyck paths (Q1347982)

Dyck paths are underdiagonal paths in the \(Z^2\) lattice, starting at the origin, never going above the main diagonal, and making east \(=(1,0)\) and north \(=(0,1)\) steps. A basic counting tool for the characteristics of Dyck paths are Catalan numbers. It turns out that a Dyck path corresponds to a word in a language generated by a specified grammar and this allows one to apply classical methods to find proper generating functions. The authors combine this approach with the Lagrange inversion formula to obtain generating function identities and further to establish unimodality properties and asymptotics of certain characteristics of Dyck paths. The main role in their study plays the so-called trinomial statistic, counting Dyck words with semilength \(n\) (i.e. Dyck paths arriving at point \((n,n\))) having \(k\) occurrences of the string 010.