The equidistribution of some Mahonian statistics over permutations avoiding a pattern of length three

Abstract

We prove the equidistribution of several multistatistics over some classes of permutations avoiding a $3$-length pattern. We deduce the equidistribution, on the one hand of inv and foze" statistics, and on the other hand that of maj and makl statistics, over these classes of pattern avoiding permutations. Here inv and maj are the celebrated Mahonian statistics, foze" is one of the statistics defined in terms of generalized patterns in the 2000 pioneering paper of Babson and Steingr\'imsson, and makl is one of the statistics defined by Clarke, Steingr\'imsson and Zeng in 1997. These results solve several conjectures posed by Amini in 2018.