Bijections for Entringer families

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 $\{1, 2, \dots, n\}$ is equal to the Euler number $E_n$. 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 $(E_{n,k})$ for computing the Euler numbers. In a series of papers, using generating function method and induction, Poupard gave several further combinatorial interpretations for $E_{n,k}$ both in alternating permutations and increasing trees. Kuznetsov, Pak, and Postnikov have given more combinatorial interpretations of $E_{n,k}$ in the model of trees. The aim of this paper is to provide bijections between the different models for $E_{n,k}$ as well as some new interpretations. In particular, we give the first explicit one-to-one correspondence between Entringer's alternating permutation model and Poupard's increasing tree model.