Factors of the Gaussian Coefficients

William Y. C. Chen, Qing-Hu Hou and Yan-Ping Mu

  Abstract:   There are many reasons for the Gaussian coefficients (or the q-binomial coefficients) to be polynomials. We show that the divisibility directly follows from the basic properties of cyclotomic polynomials. Writing the Gaussian coefficient with numerator n and denominator k in a form such that 2k ≤ n by the symmetry in k, we nd the coefficient has exactly k factors if one carries out the divisibility computation without further factorization (or as done by Maple). We further deduce the fact that the Gaussian coefficents have no multiple roots. For the n-th q-Catalan number, we show that it has exactly n -1factors after the divisibility computation.

  AMS Classification:  05A10, 33D05, 12D05.

  Suggested Running Title:   Factors of the Gaussian Coefficients. 

  Keywords:
  q-multinomial coefficient, Gaussian coefficient, q-Catalan number, cyclotomic polynomial.

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