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Let S_n(321) (respectively, S_n(132)) denote the set of all permutations of{1,2,...,n} that avoid the pattern 321 (respectively, the pattern 132).Elizalde and Pak gave a bijection Theta from S_n(321) to S_n(132) thatpreserves the numbers of fixed points and excedances for each element ofS_n(321), and commutes with the operation of taking inverses. Bloom andSaracino proved that another bijection Gamma from S_n(321) to S_n(132),introduced by Robertson, has the same properties, and they later gave apictorial reformulation of Gamma that made these results more transparent. Herewe give a pictorial reformulation of Theta, from which it follows that,although the original definitions of Theta and Gamma are very different, thesetwo bijections are in fact related to each other in a very simple way, by usinginversion, reversal, and complementation.

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