Combinatorial interpretations of the q-Faulhaber and q-Salié coefficients

Recently, Guo and Zeng discovered two families of polynomials featuring in a q-analogue of Faulhaberʹs formula for the sums of powers and a q-analogue of Gessel–Viennotʹs formula involving Saliéʹs coefficients for the alternating sums of powers. In this paper, we show that these are polynomials with symmetric, nonnegative integral coefficients by refining Gessel–Viennotʹs combinatorial interpretations.