Han's Bijection via Permutation Codes
William Y.C. Chen, Neil J.Y. Fan, and Teresa X.S. Li
Abstract: We show that Han’s bijection when restricted to permutations can be carried out in terms of the cyclic major code and the cyclic inversion code. In other words, it maps a permutation π with cyclic major code (s1, s2,..., sn) to a permutation σ with cyclic 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 partial Foata maps introduced by Björner and Wachs.
AMS Classification: 05A05, 05A15, 05A19
Keywords: Foata's second transformation, Mahonian statistic, cyclic major code, cyclic inversion code, partial Foata map