Set-valued sorting index and joint equidistributions

Abstract

Recently Petersen defined a new Mahonian index sor over the symmetric group $\mathfrak{S}_n$ and proved that $(\text{inv}, \text{rlmin})$ and $(\text{sor}, \text{cyc})$ have the same joint distribution. Foata and Han proved that the pairs of set-valued statistics $(\text{Cyc}, \text{Rmil}), (\text{Cyc}, \text{Lmap}), (\text{Rmil}, \text{Lmap})$ have the same joint distribution over $\mathfrak{S}_n$. In this paper we introduce the set-valued statistics $\text{Inv}, \text{Lmil}, \text{Sor}$ and $\text{Lmicycl}_1$ and generalize simultaneously results of Petersen and Foata-Han and find many equidistributed triples of set-valued statistics and quadruples of statistics.