论文标题
关于下降和下降的联合分布的组合证明
Combinatorial proofs on the joint distribution of descents and inverse descents
论文作者
论文摘要
令$ a_ {n，i，j} $为$（i-1）$ descents和$（j-1）$ deScents.carlitz，roselle和scoville的$ [n] $的排列数，首先揭示了一些组合和arithatorial和arithmetial的属性和$ a_ a_ a _ {盒子里的球的想法，彼得森对$ a_ {n，i，j} $的生成功能进行了组合解释，并从其生成函数中获得了$ a_ {n，i，j} $相同的复发性。 the recurrence of $A_{n,i,j}$.Let $I_{n,k}$ and $J_{n,k}$ be the number of involutions and fixed-point free involutions on $[n]$ with $k$ descents,respectively.With the help of algebraic method on generating functions,Guo and Zeng derived two recurrences of $I_{n,k}$ and $ j_ {2n，k} $在证明其非模式的属性中起着至关重要的作用。从而，发现$ a_ {n，i，j} $复发的建设性方法可以促进对这两种$ i_ {n，k} $和$ j_和j_ j_和j_ j_ j_ {2n，k k} $ i_ i_ {n，k} $ a_ {n，j} $的组合解释。
Let $A_{n,i,j}$ be the number of permutations on $[n]$ with $(i-1)$ descents and $(j-1)$ inverse descents.Carlitz, Roselle and Scoville in 1966 first revealed some combinatorial and arithmetic properties of $A_{n,i,j}$,which contain a recurrence of $A_{n,i,j}$.Using the idea of balls in boxes,Petersen gave a combinatorial interpretation for the generating function of $A_{n,i,j}$,and obtained the same recurrence of $A_{n,i,j}$ from its generating function.Subsequently, Petersen asked whether there is a visual way to understand this recurrence.In this paper,after observing the internal structures of permutation grids,we present a combinatorial proof for the recurrence of $A_{n,i,j}$.Let $I_{n,k}$ and $J_{n,k}$ be the number of involutions and fixed-point free involutions on $[n]$ with $k$ descents,respectively.With the help of algebraic method on generating functions,Guo and Zeng derived two recurrences of $I_{n,k}$ and $J_{2n,k}$ that play an essential role in the proof of their unimodal properties.Surprisingly,the constructive approach to the recurrence of $A_{n,i,j}$ is found to fuel the combinatorial interpretations of these two recurrences of $I_{n,k}$ and $J_{2n,k}$.
