A GENERALIZATION OF EULER NUMBERS TO FINITE COXETER GROUPS

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. 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. 1

Anglo-Russian Relations during the First English Revolution

Ata time when Anglo-Russian relations show a great difficulty in being re-established after the Russian revolution of 1917, it may be of special interest to look back to a very far past in English history, when England, after having fought for freedom under the leadership of Parliament against the Stuart despotism, had created by a bloody act a republic, which was taken by Protector Cromwell into his strong hands and existed for years, until death removed his grip. Then another Charles Stuart was restored to the throne which his father had left for the scaffold.</jats:p

La femme dans la Grèce antique /

Bibliogr. p. 179-183. Index