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 nâ€™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 inâˆ‡(en), whereâˆ‡ isâ€¦ (More)

Throughout this book, we abbreviate left-hand-side and right-hand-side by LHS and RHS, respectively. We let [n] = {1, . . . , n}. We let N = {0, 1, 2, . . .} denote the natural numbers, P = {1, 2, . . .} denote the positive integers, Z = {0,Â±1,Â±2, . . .} denote the integers, C denote the complex numbers, and R denote the real numbers. For any statement A,â€¦ (More)