The binomial Eulerian polynomials, first introduced in work of Postnikov, Reiner and Williams, are γ-positive polynomials and can be interpreted as h-polynomials of certain flag simplicial polytopes. Recently, Athanasiadis studied analogs of these polynomials for colored permutations and proved that they can be written as the sums of two γ-positive polynomials. In this paper, we find combinatorial interpretations of Athanasiadis' γ-positive polynomials, which leads to an alternative proof of their γ-positivity expansions using the method of group actions on colored permutations. Two results are presented. The first one is to give the γ-coefficients of the symmetric decompositions of binomial Eulerian polynomials for colored permutations, which answers a problem of Athanasiadis (2020) [3]. The second one is to give a direct combinatorial proof on the γ-expansions in the symmetric decompositions of colored derangement polynomials, which answers another problem asked by Athanasiadis (2018) [2].

在Postnikov，Reiner和Williams的工作中首次引入的二项式欧拉多项式是γ-正多项式，可以将其解释为某些标志单形多边形的h-多项式。最近，Athanasiadis研究了这些多项式的有色置换的类似物，并证明它们可以写成两个γ阳性多项式之和。在本文中，我们找到了Athanasiadis的γ阳性多项式的组合解释，这导致了使用有色置换的群作用方法证明其γ阳性展开的另一种证明。提出了两个结果。第一个是给γ二项式欧拉多项式对称分解的有色置换系数，回答了Athanasiadis（2020）的问题[3]。第二个是对有色排列多项式的对称分解中的γ展开给出直接的组合证明，这回答了Athanasiadis（2018）提出的另一个问题[2]。