  One of the most central result in combinatorics says that the descent
statistic and the excedance statistic are equidistribued over the symmetric
group. As a continuation of the work of Shareshian-Wachs (Adv. Math., 225(6)
(2010), 2921--2966), we provide a curious $t$-symmetric decomposition for the
generating polynomial of the joint distribution of the descent and excedance
statistics over the symmetric group.
