Permutation statistics and linear extensions of posets

Abstract

We consider subsets of the symmetric group for which the inversion index and major index are equally distributed. Our results extend and unify results of MacMahon, Foata, and Schützenberger, and the authors. The sets of permutations under study here arise as linear extensions of labeled posets, and more generally as order closed subsets of a partial ordering of the symmetric group called the weak order. For naturally labeled posets, we completely characterize. as postorder labeled forests, those posets whose linear extension set is equidistributed. A bijection of Foata which takes major index to inversion index plays a fundamental role in our study of equidistributed classes of permutations. We also explore, here, classes of permutations which are invariant under Foata's bijection. © 1991.