Arithmetic properties of powers of Gaussian polynomials

Everyone knows the binomial coefficients "n choose k" = n! / ( k! (n-k)! ) for 0 <= k <= n . If we replace each integer r by the polynomial 1 + q + q^2 + ... + q^{r-1} in the fomula, we obtain Gaussian polynomials, or q-binomial coefficients.  With the help of computers, the students will search patterns among coefficients of Gaussian polynomials, and their powers.  The more eager students will learn about cylindric partitions, and put their findings into use in that context.  The even more eager student will learn about q-series, and try to prove any empirical result.