Weights of permutations were originally introduced by Dugan et al. (Journal of Combinatorial Theory, Series A 164:24–49, 2019) in their study of the combinatorics of tiered trees. Given a permutation \(\sigma \) viewed as a sequence of integers, computing the weight of \(\sigma \) involves recursively counting descents of certain subpermutations of \(\sigma \). Using this weight function, one can define a q-analog \(E_n(x,q)\) of the Eulerian polynomials. We prove two main results regarding weights of permutations and the polynomials \(E_n(x,q)\). First, we show that the coefficients of \(E_n(x, q)\) stabilize as n goes to infinity, which was conjectured by Dugan et al. (Journal of Combinatorial Theory, Series A 164:24–49, 2019), and enables the definition of the formal power series \(W_d(t)\), which has interesting combinatorial properties. Second, we derive a recurrence relation for \(E_n(x, q)\), similar to the known recurrence for the classical Eulerian polynomials \(A_n(x)\). Finally, we give a recursive formula for the numbers of certain integer partitions and, from this, conjecture a recursive formula for the stabilized coefficients mentioned above.

中文翻译：

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关于置换权重和q-欧拉多项式

排列权重最初是由Dugan等人引入的。（Journal of Combinatorial Theory，Series A 164：24–49，2019）研究分层树的组合。给定一个置换\（\ sigma \）作为整数序列，计算\（\ sigma \）的权重涉及递归计算\（\ sigma \）某些子置换的下降。使用该加权函数，可以定义欧拉多项式的q模拟\（E_n（x，q）\）。我们证明了有关置换权重和多项式\（E_n（x，q）\）的两个主要结果。首先，我们证明\（E_n（x，q）\）的系数稳定为n到达无限，这是由Dugan等人推测的。（Journal of Combinatorial Theory，系列A 164：24-49，2019），并启用了具有幂等组合特性的形式幂级数\（W_d（t）\）的定义。其次，我们得出\（E_n（x，q）\）的递归关系，类似于经典欧拉多项式\（A_n（x）\）的已知递归。最后，我们给出了某些整数分区数的递归公式，并据此推测出上述稳定系数的递归公式。