论文标题
混合欧拉数字
Remixed Eulerian numbers
论文作者
论文摘要
混合的欧拉数字是尼科夫混合欧拉数字的多项式$ q $。它们在作者先前的工作中自然而然地介绍了有关定位的品种，并包含了众所周知的多项式家庭，例如$ q $ binmial系数和garsia- remmel的$ q $ thit数字。我们更深入地研究了他们的组合。作为$ Q $的多项式，它们被证明是对称的和单峰的。通过将它们解释为在简单的概率过程中计算成功概率的计算概率，我们得出了涉及加权树的组合解释。通过将固定体分解为某些组合立方体，我们获得了第二个组合解释。在$ q = 1 $的情况下，前者恢复了后尼科夫的解释，而后者恢复了刘的解释，这两者都是通过与我们不同的方法获得的。
Remixed Eulerian numbers are a polynomial $q$-deformation of Postnikov's mixed Eulerian numbers. They arose naturally in previous work by the authors concerning the permutahedral variety and subsume well-known families of polynomials such as $q$-binomial coefficients and Garsia--Remmel's $q$-hit numbers. We study their combinatorics in more depth. As polynomials in $q$, they are shown to be symmetric and unimodal. By interpreting them as computing success probabilities in a simple probabilistic process we arrive at a combinatorial interpretation involving weighted trees. By decomposing the permutahedron into certain combinatorial cubes, we obtain a second combinatorial interpretation. At $q=1$, the former recovers Postnikov's interpretation whereas the latter recovers Liu's interpretation, both of which were obtained via methods different from ours.
