On Two Bijections from \(S_n(321)\) to \(S_n(132)\)

Dan Saracino1
1Colgate University

Abstract

In \([4]\), Elizalde and Pak gave a bijection \(\Theta: S_n(321) \to S_n(132)\) that commutes with the operation of taking inverses and preserves the numbers of fixed points and excedances for every \(\Gamma \in S_n(321)\). In \([1]\) it was shown that another bijection \(\Gamma: S_n(321) \to S_n(132)\) introduced by Robertson in \([7]\) has these same properties, and in \([2]\) a pictorial reformulation of \(\Gamma\) was given that made it clearer why \(\Gamma\) has these properties. Our purpose here is to give a similar pictorial reformulation of \(\Theta\), from which it follows that, although the original definitions of \(\Theta\) and \(\Gamma\) make them appear quite different, these two bijections are in fact related to each other in a very simple way, by using inversion, reversal, and complementation.