论文标题
关于与兼容兼容的置换统计的猜想
On a conjecture concerning the shuffle-compatible permutation statistics
论文作者
论文摘要
斯坦利（Stanley）在P分区的工作中隐含了兼容兼容置换统计的概念，首先是Gessel和Zhuang明确研究的。本文的目的是证明三重$ {\ rm（udr，pk，des）} $由Gessel和Zhuang猜想，其中$ {\ rm udr} $表示上下运行的数量，$ {\ rm pk} $ DENOTE $ DENOTES $ DENOTES $ DENOTES $ deNotes $ nork norme nork nork nork nork nork nork norme nork norme nork norme nork nork norme nork norme norme normed and $ $ {这是通过建立$ {\ rm（udr，pk，des）} $来实现的 - 以贝克·贾维斯（Baker-Jarvis）和萨根（Sagan）的圣经证明了置换统计的兼容性属性。 As an application, our bijection also enables us to prove that the pair $({\rm cpk}, {\rm cdes})$ is cyclic shuffle-compatible, where ${\rm cpk}$ denotes the cyclic peak number and ${\rm cdes}$ denotes the cyclic descent number.
The notion of shuffle-compatible permutation statistics was implicit in Stanley's work on P-partitions and was first explicitly studied by Gessel and Zhuang. The aim of this paper is to prove that the triple ${\rm (udr, pk, des)}$ is shuffle-compatible as conjectured by Gessel and Zhuang, where ${\rm udr}$ denotes the number of up-down runs, ${\rm pk}$ denotes the peak number, and ${\rm des}$ denotes the descent number. This is accomplished by establishing an ${\rm (udr, pk, des)}$-preserving bijection in the spirit of Baker-Jarvis and Sagan's bijective proofs of shuffle-compatibility property of permutation statistics. As an application, our bijection also enables us to prove that the pair $({\rm cpk}, {\rm cdes})$ is cyclic shuffle-compatible, where ${\rm cpk}$ denotes the cyclic peak number and ${\rm cdes}$ denotes the cyclic descent number.
