The signed Eulerian numbers on involutions
  We define an analogue of signed Eulerian numbers $f_{n,k}$ for involutions of the symmetric group and derive some combinatorial properties of this sequence. In particular, we exhibit both an explicit formula and a recurrence for $f_{n,k}$ arising from the properties of its generating function. 
M.Barnabei, F.Bonetti, and M.Silimbani