A generalization of Euler numbers to finite Coxeter groups

Matthieu Josuat-Vergès (1, 2)

1 LIGM - Laboratoire d'Informatique Gaspard-Monge
2 UPEM - Université Paris-Est Marne-la-Vallée

It is known that Euler numbers, defined as the Taylor coefficients of the tangent and secant functions, count alternating permutations in the symmetric group. Springer defined a generalization of these numbers for each finite Coxeter group by considering the largest descent class, and computed the value in each case of the classification. We consider here another generalization of Euler numbers for finite Coxeter groups, building on Stanley's result about the number of orbits of maximal chains of set partitions. We present a method to compute these integers and obtain the value in each case of the classification. In the second part of this work, we consider maximal chains of noncrossing partitions, and how this set is divided into classes via the action of the group. We introduce a statistic related with the notion of interval partition, and show that the generating functions of classes, as well as the full generating function, are simple products. We recover Postnikov's hook-length formula in type A and obtain a variant in type B.

Mots clés: Coxeter groups; set partitions; noncrossing partitions; hook formulas

Domaines: Combinatoire [math.CO]

Matthieu Josuat-Vergès. A generalization of Euler numbers to finite Coxeter groups. Annals of Combinatorics, 2015, 19 (2), pp.325-336. ⟨10.1007/s00026-015-0267-8⟩. ⟨hal-00824258⟩