We define two symmetric $q,t$-Catalan polynomials in terms of the area and depth statistic and in terms of the dinv and dinv of depth statistics. We prove symmetry using an involution on plane trees. The same involution proves symmetry of the Tutte polynomials. We also provide a combinatorial proof of a remark by Garsia et al. regarding parking functions and the number of connected graphs on a fixed number of vertices.
@article{10_37236_10743,
author = {Joseph Pappe and Digjoy Paul and Anne Schilling},
title = {An area-depth symmetric \(q, {t\)-Catalan} polynomial},
journal = {The electronic journal of combinatorics},
year = {2022},
volume = {29},
number = {2},
doi = {10.37236/10743},
zbl = {1487.05271},
url = {http://geodesic.mathdoc.fr/articles/10.37236/10743/}
}
TY - JOUR
AU - Joseph Pappe
AU - Digjoy Paul
AU - Anne Schilling
TI - An area-depth symmetric \(q, t\)-Catalan polynomial
JO - The electronic journal of combinatorics
PY - 2022
VL - 29
IS - 2
UR - http://geodesic.mathdoc.fr/articles/10.37236/10743/
DO - 10.37236/10743
ID - 10_37236_10743
ER -
%0 Journal Article
%A Joseph Pappe
%A Digjoy Paul
%A Anne Schilling
%T An area-depth symmetric \(q, t\)-Catalan polynomial
%J The electronic journal of combinatorics
%D 2022
%V 29
%N 2
%U http://geodesic.mathdoc.fr/articles/10.37236/10743/
%R 10.37236/10743
%F 10_37236_10743
Joseph Pappe; Digjoy Paul; Anne Schilling. An area-depth symmetric \(q, t\)-Catalan polynomial. The electronic journal of combinatorics, Tome 29 (2022) no. 2. doi: 10.37236/10743
