Bijections for Entringer families
Yoann Gelineau (ICJ), Heesung Shin (ICJ), Jiang Zeng (ICJ)

TL;DR
This paper establishes explicit bijections between various combinatorial models for Entringer numbers, including alternating permutations and increasing trees, providing new interpretations and connecting existing models.
Contribution
It introduces the first explicit bijection between Entringer's alternating permutation model and Poupard's increasing tree model, unifying different combinatorial interpretations.
Findings
Established a bijection between alternating permutations and increasing trees.
Provided new combinatorial interpretations for Entringer numbers.
Connected various existing models through explicit bijections.
Abstract
Andr\'e proved that the number of alternating permutations on is equal to the Euler number . A refinement of Andr\'e's result was given by Entringer, who proved that counting alternating permutations according to the first element gives rise to Seidel's triangle for computing the Euler numbers. In a series of papers, using generating function method and induction, Poupard gave several further combinatorial interpretations for both in alternating permutations and increasing trees. Kuznetsov, Pak, and Postnikov have given more combinatorial interpretations of in the model of trees. The aim of this paper is to provide bijections between the different models for as well as some new interpretations. In particular, we give the first explicit one-to-one correspondence between Entringer's alternating permutation model and…
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