Inverses of words and the parabolic structure of the symmetric group (Q5952153)

The question of extending the notion of the inverse of a permutation to the inverse of a word (= a permutation of a multiset) was raised by Knuth. A possible approach to the problem has been given by \textit{D. Foata} and \textit{G.-N. Han} [Sémin. Lothar. Comb. 39, B39d (1997; Zbl 0886.05010)]. The authors propose another approach which meets more requirements for an ideal inversion formulated by Foata and Han.