An areadepth symmetric $q,t$Catalan polynomial
Abstract
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.
 Publication:

arXiv eprints
 Pub Date:
 September 2021
 arXiv:
 arXiv:2109.06300
 Bibcode:
 2021arXiv210906300P
 Keywords:

 Mathematics  Combinatorics;
 Primary 05A19;
 05E10;
 Secondary 05C05;
 05C30
 EPrint:
 17 pages