Eulerian polynomials of spherical type

Arjeh Cohen

The Eulerian polynomial of a finite Coxeter system (W,S) of rank n records, for each 1 ≤ k ≤ n, the number of elements w in W with an ascent set {s in S | l(vs) > l(w)} of size k, where l(w) denotes the length of w with respect to S. The classical Eulerian polynomial occurs when the Coxeter group has type An, so W is the group of all permutations on n+1 letters. In this case, it records the ascent statistics, that is, for each 1 ≤ k ≤ n, the number of permutations w on n+1 letters such that w(k) < w(k+1).

Victor Reiner gave a formula for arbitrary Eulerian polynomials and showed how to compute them in the classical cases. In this lecture, we show how to compute the Eulerian polynomial for any spherical type.

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