A proof of the q, t-square conjecture

1. the weighted sum of all Dyck paths of order n, weighted by area and bounce score; 2. the Hilbert series of the module of diagonal harmonic alternants of order n; 3. the n’th Garsia-Haiman q, t-Catalan number, which is a certain sum of complicated rational functions constructed from partitions; 4. the coefficient of the sign character in∇(en), where∇ is the Bergeron-Garsia nabla operator [1, 10], and en is an elementary symmetric function. Precise definitions of the terms mentioned here will be given later (§2). Loehr and Warrington [12] recently found an analogue of this theorem that involves q, t-analogues of lattice paths inside squares. Their result, which we call the q, t-square conjecture, states that the following five quantities are equal for every n: