Bijections for Entringer families

André proved that the number of down–up permutations on { 1 , 2 , … , n } is equal to the Euler number E n . 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 ( E n , 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 E n , k both in down–up permutations and for 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 down–up permutation model and Poupard’s increasing tree model.