Sách A recurrence relation for the “inv” analogue of q-Eulerian polynomials

The above q-recurrence relation is “semi” in the sense that the summands on the right
involve two factors one of which depends on q whereas the other does not. We shall
establish in the present note that a “fully” q-recurrence relation for A
des,inv
n
(t, q) exists
such that both factors of the summands depend on q (see Theorem 2.2 below). In the
next section, we derive this recurrence relation algebraically. In the final section, we give
a combinatorial proof of this recurrence relation.