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Bijections around Springer numbers

Shaoshi Chen, Yang Li, Zhicong Lin, Sherry H. F. Yan

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.

Bijections around Springer numbers

Abstract

Arnol'd proved in 1992 that Springer numbers enumerate the Snakes, which are type 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 -dimensional permutations are also counted by Springer numbers. In this work, we construct a sequence of ``natural'' bijections linking the above four combinatorial objects.
Paper Structure (4 sections, 6 theorems, 24 equations, 1 figure)

This paper contains 4 sections, 6 theorems, 24 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Bijecting weakly increasing $3$-dimensional permutations with snakes
  3. Bijecting snakes with labeled ballot paths
  4. Concluding remarks

Key Result

Theorem 2.1

The mapping $\Phi: \mathfrak{W}_n\rightarrow\mathcal{S}_n$ is a bijection.

Figures (1)

  • Figure 1: Bijections around Springer numbers

Theorems & Definitions (10)

  • Theorem 2.1
  • proof
  • Lemma 3.1
  • proof
  • Theorem 3.2
  • proof
  • Lemma 3.3
  • Lemma 3.4
  • proof
  • Theorem 3.5