The descent statistic on involutions is not log-concave

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.