
Permutations and the Rest. Algebraic Combinatorics of Permutations. 295
sight. That was corrected in [34]. The proof we present here is significantly
simpler than the previous proof.
PROOF (of Theorem 7.32) Let p ∈ P
A
n
. Recall Exercise 27 of Chapter 4.
Define f (p) to be the 132-avoiding permutation whose unlabeled binary tree is
obtained from the unlabeled binary tree T (p)ofp by reflecting T (p) through
a vertical axis. By part (b) of the mentioned exercise, this reflection will turn
left edges into right edges, and so ascents into descents, and vice versa. In
particular, the vertex that was in the ith position from the left will now be
in the ith position ...