Bijections for Entringer families

André proved that the number of down–up permutations on {1,2,…,n}{1,2,…,n} is equal to the Euler number EnEn. A refinement of André’s result was given by Entringer, who proved that counting down–up permutations according to the first element gives rise to Seidel’s triangle (En,k)(En,k) for computing the Euler numbers. In a series of papers, using the generating function method and induction, Poupard gave several further combinatorial interpretations for En,kEn,k both in down–up permutations and for increasing trees. Kuznetsov, Pak, and Postnikov have given more combinatorial interpretations of En,kEn,k in the model of trees. The aim of this paper is to provide bijections between the different models for En,kEn,k as well as some new interpretations. In particular, we give the first explicit one-to-one correspondence between Entringer’s down–up permutation model and Poupard’s increasing tree model.