On a conjecture concerning the shuffle-compatible permutation statistics

Abstract

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 $(\mathrm{udr},\mathrm{pk},\mathrm{des})$ is shuffle-compatible as conjectured by Gessel and Zhuang, where $\mathrm{udr}$ denotes the number of up-down runs, $\mathrm{pk}$ denotes the peak number, and $\mathrm{des}$ denotes the descent number. This is accomplished by establishing an $(\mathrm{udr},\mathrm{pk},\mathrm{des})$-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 $(\mathrm{cpk},\mathrm{cdes})$ is cyclic shuffle-compatible, where $\mathrm{cpk}$ denotes the cyclic peak number and $\mathrm{cdes}$ denotes the cyclic descent number.