The γ-positivity of basic Eulerian polynomials via group actions

We provide combinatorial interpretation for the γ-coefficients of the basic Eulerian polynomials that enumerate permutations by the excedance statistic and the major index as well as the corresponding γ-coefficients for derangements. Our results refine the classical γ-positivity results for the Eulerian polynomials and the derangement polynomials. The main tools are Branden's modified Foata-Strehl action on permutations and the recent triple statistic ( des , rix , aid ) equidistributed with ( exc , fix , maj ).

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