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 2kn 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

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