Investigating the Symmetry of the q,t-Catalan Polynomials Using New Statistics on Plane Binary Trees, Triangulations of Convex Polygons, and Paired Lattice Paths

Abstract

There exist polynomials Cn(q,t) known as the q,t-Catalan polynomials. We know from work in representation theory by Haglund, Haiman, and Garsia that the q,t-Catalan polynomials are symmetric; that is, that Cn(q,t) = Cn(t,q). The q,t-Catalan polynomials are known combinatorially as the weighted sums of Dyck paths. A Dyck path is a lattice path in the xy-plane which begins at the origin and ends at the point (n,n) by way of moving a distance of one incrementally by either moving one unit to the right or one unit up, while staying below the diagonal line y=x. The number of Dyck paths of order n is given by the Catalan number Cn. We can find the statistics area, dinv, and bounce on Dyck paths such that for Cn(q,t) the q keeps track of the area of the Dyck paths and the t keeps track of either bounce or dinv. Even though we know that Cn(q,t) is symmetric, we have no combinatorial explanation for why this is so. We will look at three other combinatorial objects: plane binary trees, triangulations of convex polygons, and paired lattice paths; all of which are counted by the Catalan numbers. New statistics will be introduced on these objects in hopes of developing a combinatorial reason for the symmetry of the q,t-Catalan polynomials.