The descent statistic is despi despi the descent statistic

Question

Math 121c: Topics in Geometric Combinatorics, Spring 2012 Problems-

For π ∈ S_n, the descent set of π is

Des(π) := {j ∈ [n - 1] : π(j) > π(j + 1)}.

The descent statistic is des(π) = |Des(π)|. The descent statistic is encoded in the Eulerian polynomial ∑_π∈Sn t^des(π), and the most basic identity for Eulerian polynomials is

∑_k≥0(k + 1)ⁿt^k = ∑_{π∈S_n} t^des(π)/(1 - t)ⁿ⁺¹.

Prove this by proving the following more general result:

∑_k≥0j=1∏ⁿ[k + 1]z_jz₀^k = ∑_{π∈S_n}(∏_j∈Des(π)z₀z_π(1)z_π(2)· · · z_π(j)/_j=0∏ⁿ(1 - z₀z_π(1)z_π(2)· · · z_π(j))),

where [m]_q = 1 + q + q² + · · · + q^m-1.

(Hint: Integer point transform on the cone over [0, 1]ⁿ, together with triangulation.)