The Eulerian distribution on involutions is indeed unimodal

Let I n , k (respectively J n , k ) be the number of involutions (respectively fixed-point free involutions) of { 1 , … , n } with k descents. Motivated by Brentiʹs conjecture which states that the sequence I n , 0 , I n , 1 , … , I n , n − 1 is log-concave, we prove that the two sequences I n , k and J 2 n , k are unimodal in k, for all n. Furthermore, we conjecture that there are nonnegative integers a n , k such that ∑ k = 0 n − 1 I n , k t k = ∑ k = 0 ⌊ ( n − 1 ) / 2 ⌋ a n , k t k ( 1 + t ) n − 2 k − 1 . This statement is stronger than the unimodality of I n , k but is also interesting in its own right.