The descent statistic on involutions is not log-concave

We establish a combinatorial connection between the sequence (in,k)(in,k) counting the involutions on nn letters with kk descents and the sequence (an,k)(an,k) enumerating the semistandard Young tableaux on nn cells with kk symbols. This allows us to show that the sequences (in,k)(in,k) are not log-concave for some values of nn, hence answering a conjecture due to F. Brenti.