THESIS
2016
ix, 56 pages : illustrations ; 30 cm
Abstract
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
nk to itself, where k = 0,1 and P
nk = {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...[
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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
nk to itself, where k = 0,1 and P
nk = {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.
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