Some simple bijections involving lattice walks and ballot sequences

2010/10/23 by Marc A. A. Van Leeuwen, Marc Van Leeuwen, Van Leeuwen, Marc A. A. · 1 citation
Mathematics · #Advanced Combinatorial Mathematics #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Combinatorics (math.CO) #FOS: Mathematics #math.CO

paper · pdf · doi:10.48550/arxiv.1010.4847

arxiv created 2010/10/23 · openalex publication_date 2010/10/23 · arxiv updated 2010/10/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this note we observe that a bijection related to Littelmann's root operators (for type A₁) transparently explains the well known enumeration by length of walks on \N (left factors of Dyck paths), as well as some other enumerative coincidences. We indicate a relation with bijective solutions of Bertrand's ballot problem: those can be mechanically transformed into bijective proofs of the mentioned enumeration formula.

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