Bijections for Entringer families

Yoann Gelineau (1) , Heesung Shin (1) , Jiang Zeng (1)

1 ICJ - Institut Camille Jordan

The Seidel-Entringer triangle is a double indexed sequence $(E_{n,k})$ refining the Euler numbers, whose combinatorial interpretation in alternating permutations was first given by Andr\'{e}. A refinement of Andr\'{e}'s interpretation for $E_{n,k}$ was given by Entringer, who proved that these numbers count alternating permutations according to the first element. In a series of papers, Poupard provided more combinatorial interpretations for $E_{n,k}$ by analytic methods or induction. The aim of this paper is to provide bijections between the different models for $E_{n,k}$. In particular, we establish the first one-to-one correspondence between Entringer's alternating permutations model and Poupard's 0-1-2 increasing trees model.

Mots clés: Entringer numbers; Euler numbers; alternating permutations; 0-1-2 increasing words; direct alternating permutations; G-words; R-words; U-words

Domaines: Combinatoire [math.CO]

Yoann Gelineau, Heesung Shin, Jiang Zeng. Bijections for Entringer families. 2010. ⟨hal-00472187v2⟩