Set-valued sorting index and joint equidistributions

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}$, $\text{Rmil}$), ($\text{Cyc}$, $\text{Lmap}$), ($\text{Rmil}$, $\text{Lmap}$)$havethesamejointdistributionover$\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.