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We study the descent distribution over the set of centrosymmetric permutations that avoid the pattern of length 3. Our main tool in the most puzzling case, namely, $\tau=123$ and $n$ even, is a bijection that associates a Dyck prefix of length $2n$ to every centrosymmetric permutation in $S_{2n}$ that avoids 123., Comment: 17 pages, 6 figures

The paper was withdrawn because of its significant overlap with a paper appeared recently., Comment: withdrawn paper

We establish a combinatorial connection between the sequence $(i_{n,k})$ counting the involutions on $n$ letters with $k$ descents and the sequence $(a_{n,k})$ enumerating the semistandard Young tableaux on $n$ cells with $k$ symbols. This allows us to show that the sequences $(i_{n,k})$ are not log-concave for some values of $n$, hence answering a conjecture due to F. Brenti., Comment: 10 pages