A generalization of Euler numbers to finite Coxeter groups

TL;DR

This paper introduces a new generalization of Euler numbers for finite Coxeter groups, extending classical combinatorial concepts and providing a method to compute these generalized numbers across different group types.

Contribution

It proposes a novel generalization of Euler numbers for finite Coxeter groups based on Stanley's orbit counting, with a computational method for all cases.

Findings

Derived explicit formulas for the new Euler numbers in each Coxeter group type.

Established a computational approach to determine these numbers systematically.

Extended classical permutation enumeration to a broader algebraic context.

Abstract

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.