2024/12/31 by Shaoshi Chen, Chen, Shaoshi, Li Yang +5
Mathematics · #Advanced Topics in Algebra #Combinatorics (math.CO) #FOS: Mathematics #Graph theory and applications #Mathematics and Applications
paper · pdf · doi:10.48550/arxiv.2501.00335
openalex publication_date 2024/12/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Arnol'd proved in 1992 that Springer numbers enumerate the Snakes, which are type B analogs of alternating permutations. Chen, Fan and Jia in 2011 introduced the labeled ballot paths and established a ``hard'' bijection with snakes. Callan conjectured in 2012 and Han--Kitaev--Zhang proved recently that rc-invariant alternating permutations are counted by Springer numbers. Very recently, Chen--Fang--Kitaev--Zhang investigated multi-dimensional permutations and proved that weakly increasing 3-dimensional permutations are also counted by Springer numbers. In this work, we construct a sequence of ``natural'' bijections linking the above four combinatorial objects.