Factors of the Gaussian Coefficients
William Y.C. Chen and Qing-Hu Hou
Abstract: We present some simple observations on factors of the q-binomial coefficients, the q-Catalan numbers, and the q-multinomial coefficients. Writing the Gaussian coefficient with numerator n and denominator k in a form such that 2k≤ n by the symmetry in k, we show that this coefficient has at least k factors. Some divisibility results of Andrews, Brunetti and Del Lungo are also discussed.
AMS Classification: 05A10, 33D05,12D05
Suggested Running Title: Factors of the Gaussian Coefficient
Keywords: q-multinomial coefficient, Gaussian coefficient, q-Catalan number, cyclotomic polynomial