论文标题
组合三角形的统一方法：广义的欧拉尔多项式
A unified approach to combinatorial triangles: a generalized Eulerian polynomial
论文作者
论文摘要
由超巨大群体中的经典欧拉数，下降和脱颖而出的动机，一个来自楼梯tableaux的三角形阵列等等，我们研究了三角阵列$ [\ nathcal {\ nathcal {t} _ {n，k} _ {n，k} _ {n，k} _ { {t} _ {n，k} =λ（a_0n+a_1k+a_2）\ Mathcal {t} _ {n-1，k}+（b_0n+b_1k+b_1k+b_2）\ Mathcal \ Mathcal {t} {t} _ {n-1，k-2} \ end {equation*}，带有$ \ MATHCAL {T} _ {0,0,0} = 1 $和$ \ Mathcal {t} _ {n，k} = 0 $否则，除非$ 0 \ 0 \ 0 \ le k \ le k \ le k \ le le n $。我们为其创建函数$ \ MATHCAL {t} _n（x）$从行生成函数$ a_n（x）$的另一个数组$ [a_ {a_ {n，k}] _ {n，k} _ {n，k} $满足两项复发的关系。基于此转换，我们可以获得$ \ Mathcal {t} _ {n，k} $和$ \ Mathcal {t} _n（x）$的属性，包括非神经性，log-conconcavity，real-concovity，real lotedness，real loot rotedness，earplicit formula等。然后，我们将著名的Frobenius公式，$γ$阳性分解和用于经典欧拉尔多项式的David-Barton公式扩展到广义的Eulerian多项式。我们还获得了通用的欧拉多项式与一般衍生多项式多项式的身份。 最后，我们将结果应用于兰伯特功能的阵列，楼梯tableaux的三角形阵列以及统一方法中$ b $的交替运行三角形。
Motivated by the classical Eulerian number, descent and excedance numbers in the hyperoctahedral groups, an triangular array from staircase tableaux and so on, we study a triangular array $[\mathcal {T}_{n,k}]_{n,k\ge 0}$ satisfying the recurrence relation: \begin{equation*} \mathcal {T}_{n,k}=λ(a_0n+a_1k+a_2)\mathcal {T}_{n-1,k}+(b_0n+b_1k+b_2)\mathcal {T}_{n-1,k-1}+\frac{cd}λ(n-k+1)\mathcal {T}_{n-1,k-2} \end{equation*} with $\mathcal {T}_{0,0}=1$ and $\mathcal {T}_{n,k}=0$ unless $0\le k\le n$. We derive a functional transformation for its row-generating function $\mathcal{T}_n(x)$ from the row-generating function $A_n(x)$ of another array $[A_{n,k}]_{n,k}$ satisfying a two-term recurrence relation. Based on this transformation, we can get properties of $\mathcal {T}_{n,k}$ and $\mathcal{T}_n(x)$ including nonnegativity, log-concavity, real rootedness, explicit formula and so on. Then we extend the famous Frobenius formula, the $γ$ positivity decomposition and the David-Barton formula for the classical Eulerian polynomial to those of a generalized Eulerian polynomial. We also get an identity for the generalized Eulerian polynomial with the general derivative polynomial. Finally, we apply our results to an array from the Lambert function, a triangular array from staircase tableaux and the alternating-runs triangle of type $B$ in a unified approach.
