More bijections for Entringer and Arnold families

TL;DR

This paper establishes new bijections linking Entringer and Arnold numbers to various permutation classes and combinatorial structures, enhancing understanding of their combinatorial interpretations.

Contribution

It introduces novel bijections connecting Entringer and Arnold numbers to André, Simsun, signed André permutations, and type B increasing 1-2 trees.

Findings

Bijections for Entringer numbers in André and Simsun permutations.

Bijections for Arnold numbers in signed André permutations and type B increasing 1-2 trees.

Enhanced combinatorial understanding of these special numbers.

Abstract

The Euler number $E_n$ (resp. Entringer number $E_{n,k}$) enumerates the alternating (down-up) permutations of $\{1,\dots,n\}$ (resp. starting with $k$). The Springer number $S_n$ (resp. Arnold number $S_{n,k}$) enumerates the type $B$ alternating permutations (resp. starting with $k$). In this paper, using bijections we first derive the counterparts in {\em Andr\'e permutations} and {\em Simsun permutations} for the Entringer numbers $(E_{n,k})$, and then the counterparts in {\em signed Andr\'e permutations} and {\em type $B$ increasing 1-2 trees} for the Arnold numbers $(S_{n,k})$.