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.

12 pages, 1 figure