2022/01/18 by Lihong Yang, Yang, Lihong, Sherry H.F. Yan +1
Computer Science · Mathematics · #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Bayesian Methods and Mixture Models #Combinatorics (math.CO) #FOS: Mathematics
paper · pdf · doi:10.48550/arxiv.2201.06784
openalex publication_date 2022/01/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.