Bijections around Springer numbers

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.