An open problem about proving symmetry phenomenon of q; t-Catalan Polynomial combinatorially, was introduced by James Haglund.

Ofir Ammar has suggested a possible generalization related to the set of parking functions P_n in his master thesis in order to tackle the problem. This thesis generalizes some of Ofir Ammar's results and gives a detailed proof for the bijection from the set P_n^k to itself, where k = 0,1 and P_n^k = {p ∈ P_n : area(p) + dinv(p) = (₂ⁿ) - k}, in which the bijection swaps the area statistics with dinv statistics, and preserves the occupant(1). Also, this thesis gives some conjectures, and provides an involution for Dyck path π with area(π) ≤ 1 or dinv(π) ≤ 1, which swaps the two statistics. This thesis would lead to a possibility for further investigation of th...[ Read more ]

An open problem about proving symmetry phenomenon of q; t-Catalan Polynomial combinatorially, was introduced by James Haglund.

Ofir Ammar has suggested a possible generalization related to the set of parking functions P_n in his master thesis in order to tackle the problem. This thesis generalizes some of Ofir Ammar's results and gives a detailed proof for the bijection from the set P_n^k to itself, where k = 0,1 and P_n^k = {p ∈ P_n : area(p) + dinv(p) = (₂ⁿ) - k}, in which the bijection swaps the area statistics with dinv statistics, and preserves the occupant(1). Also, this thesis gives some conjectures, and provides an involution for Dyck path π with area(π) ≤ 1 or dinv(π) ≤ 1, which swaps the two statistics. This thesis would lead to a possibility for further investigation of the symmetry problem about the q, t-Catalan polynomials and its generalization.