Han's Bijection via Permutation Codes

TL;DR

This paper explores Han's bijection on permutations, demonstrating its relation to cyclic codes and characterizing fixed points through Foata's transformation, thus deepening understanding of permutation code mappings.

Contribution

It establishes a new interpretation of Han's bijection using cyclic major and inversion codes, linking it to Foata's transformation and fixed point characterization.

Findings

Han's bijection maps cyclic major codes to cyclic inversion codes.

Fixed points of Han's map are characterized by strong fixed points of Foata's transformation.

The work connects permutation codes with classical bijections in combinatorics.

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 $\pi$ with a cyclic major code $(s_1, s_2, ..., s_n)$ to a permutation $\sigma$ with a cyclic inversion code $(s_1,s_2, ..., s_n)$. 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 Bj\"orner and Wachs.