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arXiv · 2501.00335

Bijections around Springer numbers

Shaoshi Chen·Yang Li·Zhicong Lin·Sherry H. F. Yan
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Abstract

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.

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Shaoshi Chen, Yang Li, Zhicong Lin, Sherry H. F. Yan. 2024-12-31. Bijections around Springer numbers. https://arxiv.org/abs/2501.00335

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