Descent polynomials and peak polynomials, which enumerate permutations $\pi \in {\mathfrak{S}}_n$ with given descent and peak sets respectively, have recently received considerable attention (Billey et al., 1989;Diaz-Lopez et al., 2019). We give several formulas for $q$-analogs of these polynomials which refine the enumeration by the length of $\pi$. In the case of $q$-descent polynomials we prove that the coefficients in one basis are strongly $q$-log concave, and conjecture this property in another basis. For peaks, we prove that the $q$-peak polynomial is palindromic in $q$, resolving a conjecture of Diaz-Lopez et al. (2021).

中文翻译：

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上 $q$- 下降和峰值多项式的模拟

枚举排列的下降多项式和峰值多项式$\pi \in {\mathfrak{秒}}_n$分别具有给定的下降和峰值集，最近受到了相当大的关注（Billey 等人，1989 年；Diaz-Lopez 等人，2019 年）。我们给出了几个公式$q$- 这些多项式的模拟，通过长度细化枚举 $\pi$. 如果是$q$-下降多项式我们证明一个基中的系数是强的 $q$-log 凹，并在另一个基础上推测此属性。对于峰，我们证明$q$- 峰值多项式是回文 $q$，解决了 Diaz-Lopez 等人的猜想。(2021)。