Question: q-binomial coefficient

Problem 5.

Definition. The q-binomial coefficient ---denoted by qbinomial(n,k,q) --is defined by the formula
      qbinomial(n,k,q)    =       
"Product(q^i - 1, i = n - k + 1 .. n)/Product(q^i - 1, i = 1 .. 

   k)"

qbinomial(n,k,q) is a polynomial of degree k*(n-k) in the variable q. These polynomials are also called Gaussian polynomials or Gaussian  coefficients. They occur in various places in mathematics. For example qbinomial(n,k,q) is the number of k-dimensional subspaces of an n-dimensional vector space over a field with q elements.  An example:  qbinomial(6,3,q)  is the polynomial 

"q^9 + q^8 + 2*q^7 + 3*q^6 + 3*q^5 + 3*q^4 + 3*q^3 + 2*q^2 + q 

   + 1"

(a) Write a procedure to find qbinomial(n,k,q) and check to see that it agrees with the above for n=6  and k=3.  Note that to get nice output you will need to simplify, expand and sort the expression.

(b) Check whether or not the sequence of coefficients of the Gaussian polynomial qbinomial(n,k,q) is unimodal for  0 =< n < =10 and 0=< k < =n.  Do not print out all these in your final work. Just write a program to do the checking.  Have it print out the number of pairs (n,k) for which qbinomial(n,k,q) 's coefficients are not unimodal. This is a number from 0 to 66.

[Hint: You may use the following to convert the coefficients of a polynomial f in the variable q to a list:
>L:=[seq(coeff(f,q,i),i=0..degree(f,q))]; ]