Recently, Guo and Zeng discovered two families of polynomials featuring in a q-analogue of Faulhaber's formula for the sums of powers and a q-analogue of Gessel-Viennot's formula involving Salié's coefficients for the alternating sums of powers. In this paper, we show that these are polynomials with symmetric, nonnegative integral coefficients by refining Gessel-Viennot's combinatorial interpretations. © 2006 Elsevier Inc. All rights reserved.
Guo, V. J. W., Rubey, M., & Zeng, J. (2006). Combinatorial interpretations of the q-Faulhaber and q-Salié coefficients. Journal of Combinatorial Theory. Series A, 113(7), 1501–1515. https://doi.org/10.1016/j.jcta.2006.01.007