The Major Index Specialization of the \(q, t\)-Catalan

Abstract

Various authors have defined statistics on Dyck paths that lead to generalizations of the Catalan numbers. Three such statistics are area, maj, and bounce. Haglund, whe introduced the bounce statistic, gave an algebraic proof that \(n(n – 1)/2+\) area — bounce and maj have the same distribution on Dyck paths of order \(n\). We give an explicit bijective proof of the same result.