In [2] we proved that a natural bijection \(\Gamma:S_n(321) \rightarrow S_n(132)\) that Robertson defined by an iterative process in [8] preserves the numbers of fixed points and excedances in each \(\sigma\in S_n(321).\) The proof depended on first showing that \(\Gamma(\sigma^{-1})=(\Gamma(\sigma))^{-1}\) for all \(\sigma\in S_n(321).\)
Here we give a noniterative definition of \(\Gamma\) that frees the result about fixed points and excedances from its dependence on the result about inverses, while also greatly simplifying and elucidating the result about inverses. We also establish a simple connection between \(\Gamma\) and an analogous bijection \(\phi^*:S_n(213)\rightarrow S_n(321)\) introduced in [1] and studied in [3].
Published: Adv. in Appl. Math (45) 2010, 395-409.