Because there are only finitely many possibilities for the number of descents of an affine permutation, but infinitely many affine permutations, it is not possible to naively form a generating function for affine permutations by number of descents (an affine analogue of Eulerian polynomials). One possible resolution is to consider affine descents (equivalently, cyclic descents) in the finite symmetric group S n . Another is to consider simultaneously the length and number of descents of an affine permutation. The multivariate generating function for these statistics over S ~ ~ n simultaneously for all n is ∑ ∑ n ≥ ≥ 1 x n 1 − − q n ∑ ∑ w ∈ ∈ S ~ ~ n t des ⁡ ⁡ ( w ) q ℓ ℓ ( w ) = [ x ⋅ ⋅ ∂ ∂ ∂ ∂ x log ⁡ ⁡ ( exp ⁡ ⁡ ( x ; q ) ) 1 − − t exp ⁡ ⁡ ( x ; q ) ] x ↦ ↦ x 1 − − t 1 − − q where des(w) is the number of descents of the affine permutation w and exp ⁡ ⁡ ( x ; q ) = ∑ ∑ n ≥ ≥ 0 x n ( 1 − − q ) n ( 1 − − q ) ( 1 − − q 2 ) ⋯ ⋯ ( 1 − − q n ) is the q -exponential function.