Rational Catalan polynomials and rank words

2016/11/15 by Ryan Kaliszewski, Kaliszewski, Ryan, Huilan Li +1
Agricultural and Biological Sciences · Mathematics · #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Botanical Research and Chemistry #Combinatorics (math.CO) #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.1611.04956

openalex publication_date 2016/11/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Related

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