Proof of Dilks' bijectivity conjecture on Baxter permutations

2021/12/22 by Zhicong Lin, Jing Liu, Lin, Zhicong +1
Mathematics · #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Advanced Topics in Algebra #Combinatorics (math.CO) #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.2112.11698

openalex publication_date 2021/12/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Baxter permutations originally arose in studying common fixed points of two commuting continuous functions. In 2015, Dilks proposed a conjectured bijection between Baxter permutations and non-intersecting triples of lattice paths in terms of inverse descent bottoms, descent positions and inverse descent tops. We prove this bijectivity conjecture by investigating its connection with the Françon--Viennot bijection. As a result, we obtain a permutation interpretation of the (t,q)-analog of the Baxter numbers \frac1n+1\brack 1_qn+1\brack 2_q∑_k=0^n-1q^3k+1\choose2n+1\brack k_qn+1\brack k+1_qn+1\brack k+2_qt^k, where n\brack k_q denote the q-binomial coefficients.

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