Statistics-preserving bijections between classical and cyclic permutations

Recently, Elizalde (2011) [2] has presented a bijection between the set Cn+1Cn+1 of cyclic permutations on {1,2,…,n+1}{1,2,…,n+1} and the set of permutations on {1,2,…,n}{1,2,…,n} that preserves the descent set of the first n   entries and the set of weak excedances. In this paper, we construct a bijection from Cn+1Cn+1 to SnSn that preserves the weak excedance set and that transfers quasi-fixed points into fixed points and left-to-right maxima into themselves. This induces a bijection from the set DnDn of derangements to the set Cn+1q of cycles without quasi-fixed points that preserves the weak excedance set. Moreover, we exhibit a kind of discrete continuity between Cn+1Cn+1 and SnSn that preserves at each step the set of weak excedances. Finally, some consequences and open problems are presented.

► We study some statistics (weak excedance, fixed point, left-to-right maxima) on permutations, derangements and cyclic permutations. ► We provide several bijections that preserve weak excedance sets. ► Finally we gives two open problems for descent sets.