2020/02/28 by Nancy Wallace, Wallace, Nancy
Computer Science · Mathematics · #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Combinatorics (math.CO) #FOS: Mathematics #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.2003.00062
openalex publication_date 2020/02/28 · openalex created_date 2020/03/06 · openalex updated_date 2026/07/28
This paper contains a partial answer to the open problem 3.11 of \cite[H2008]. That is to find an explicit bijection on Schröder paths that inverts the statistics area and bounce. This paper started as an attempt to write the sum over m-Schröder paths with a fix number of diagonal steps into Schur functions in the variables q and t. Some results have been generalized to parking functions, and some bijections were made with standard Young tableaux giving way to partial combinatorial formulas in the basis sμ(q,t)sλ(X) for ∇(en) (respectively, ∇m(en)), when μ and λ are hooks (respectively, μ is of length one). We also give an explicit algorithm that gives all the Schröder paths related to a Schur function sμ(q,t) when μ is of length one. In a sense, it is a partial decomposition of Schröder paths into crystals.