Gamma Positivity of the Excedance-Based Eulerian Polynomial in Positive Elements of Classical Weyl Groups

The Eulerian polynomial An(t) enumerating descents in Sn is known to be gamma positive for all n. When enumeration is done over the type B and type D Coxeter groups, the type B and type D Eulerian polynomials are also known to be gamma positive for all n. We consider An (t) and A − n (t), the polynomials which enumerate descents in the alternating group An and in Sn − An respectively. We show the following results about An (t) and A − n (t): both polynomials are gamma positive iff n ≡ 0, 1 (mod 4). When n ≡ 2, 3 (mod 4), both polynomials are not palindromic. When n ≡ 2 (mod 4), we show that two gamma positive summands add up to give An (t) and A − n (t). When n ≡ 3 (mod 4), we show that three gamma positive summands add up to give both An (t) and A − n (t). We show similar gamma positivity results about the descent based type B and type D Eulerian polynomials when enumeration is done over the positive elements in the respective Coxeter groups. We also show that the polynomials considered in this work are unimodal. Mathematics Subject Classifications: 05A05, 05A15