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The classical Eulerian polynomials An(t) are known to be gamma-positive. Define
the positive Eulerian polynomial A_n^+(t) as the polynomial obtained when we sum descents over
the alternating group. We show that A_n^+(t) is gamma-positive if and only if n ≡ 0, 1 (mod 4).
When n ≡ 2 (mod 4), we show that A_n^+(t) can be written as a sum of two gamma-positive
polynomials while if n ≡ 3 (mod 4), we show that A_n^+(t) can be written as a sum of three gamma-
positive polynomials. Similar results are shown when we consider the positive Type B and Type D
Eulerian polynomials.
Macmahon showed that descents and excedances are equidistributed over the symmetric group
but it is easy to check that they are not equidistributed over the alternating group. We characterize
the gamma-positivity results also for the excedance based polynomials in the positive elements of
Classical Weyl Groups. |