A q-analogue of generalized Eulerian polynomials with applications

Abstract We show that counting permutations with respect to double descents, double ascents, peaks, valleys and inversions yields a natural q-analogue of a classical formula of Carlitz–Scoville and generalizes Zhuang's recent formula counting inversions, peaks, and descents of permutations [17] . We also prove a left-peak analogue of Carlitz–Scoville's formula that recovers several recent results of Zhuang [17] . In contrast to Zhuang's proof involving theory of non-commutative symmetric functions, our approach based on q-calculus is more elementary.