Bijections around Springer numbers

TL;DR

This paper constructs a sequence of natural bijections linking various combinatorial objects, including snakes, labeled ballot paths, rc-invariant alternating permutations, and multi-dimensional permutations, all counted by Springer numbers.

Contribution

It establishes explicit bijections among four different combinatorial objects counted by Springer numbers, unifying their combinatorial interpretations.

Findings

Bijections between snakes and labeled ballot paths

Bijections involving rc-invariant alternating permutations

Connections to multi-dimensional permutations counted by Springer numbers

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.