Descent sets of cyclic permutations

We present a bijection between cyclic permutations of {1,2,…,n+1} and permutations of {1,2,…,n} that preserves the descent set of the first n entries and the set of weak excedances. This non-trivial bijection involves a Foata-like transformation on the cyclic notation of the permutation, followed by certain conjugations. We also give an alternate derivation of the consequent result about the equidistribution of descent sets using work of Gessel and Reutenauer. Finally, we prove a conjecture of the author in [S. Elizalde, The number of permutations realized by a shift, SIAM J. Discrete Math. 23 (2009) 765-786] and a conjecture of Eriksen, Freij and Wästlund.