On Han’s Bijection via Permutation Codes

We show that Han’s bijection when restricted to permutations can be carried out in terms of the major code and inversion code. In other words, it maps a permutation π with a major code (s1, s2, . . . , sn) to a permutation σ with an inversion code (s1, s2, . . . , sn). We also show that the fixed points of Han’s map can be characterized by the strong fixed points of Foata’s second fundamental transformation. The notion of strong fixed points is related to the partial Foata maps introduced by Björner and Wachs. We further give a construction of a class of Mahonian statistics on permutations in terms of the major code.