Rational Catalan polynomials and rank words

TL;DR

This paper introduces new combinatorial statistics and objects, such as skip and rank words, to study rational Catalan polynomials, providing explicit formulas and symmetry properties for specific cases.

Contribution

It defines the skip statistic and rank words for rational Dyck paths, establishing their properties and connections, and derives explicit formulas for (3,n)-rational Catalan polynomials.

Findings

Defined the skip statistic for rational Dyck paths

Established a correspondence between rank words and Dyck paths

Proved (3,n)-rational Catalan polynomials are q,t-symmetric

Abstract

For $m,n$ coprime we introduce a new statistic skip on $(m,n)$-rational Dyck paths and give a fast way to compute dinv and skip statistics. We also introduce $(m,n)$-rank words, which are in one-to-one correspondence with $(m,n)$-Dyck paths. Defining an equivalence relation on pairs of certain ranks in a rank word, we prove that the number of equivalence classes is the skips of the rank word, and the skips of the corresponding Dyck path. We construct a homogeneous generating function $W_{m,n}(q,t,b)$ using statistics area, dinv and skip, where $W_{m,n}(q,t,1)=C_{m,n}(q,t)$, the rational Catalan polynomial. We then give an explicit formula for $(3,n)$-rational Catalan polynomials and prove they are $q,t$-symmetric.