I've in my Combinatorics book a short paragraph about the Rook Polynomials and the way they should be calculated, the book doesn't dive into much details and attempt to explain the concept with an exercise:
"Professor J wants to teach A or B, Professor S wants to teach B or C and Professor G wants to teach A or C. Each Professor can be assigned to teach at most one course, with no more than one Professor per course, and a Professor only gets a course that he or she wants to teach. Set up a generating function and use it to answer the following questions:
1) In how many ways can we assign one Professor to a course?
2) Same as 1 with two Professors
3) Same as 1 with three Professors."
I setup a board of $$3x3$$ elements, with the Professors on the columns and the courses on the rows, the board looks like this:
JSG
AOXO
BOOX
CXOO
Where you see J can teach A or B, S can teach B or C &c.
The generating function I found for this board seems to be $$R(x,B)=1+6x+9x^2+2x^3$$ And I see that $$R(1,B)=18$$ What does that number mean? Is 18 the number of ways to assign one Professor to a course? Of course not, because one Professor can be assigned to one of the courses in 6 ways.
Same for $$R(2,B)=65$$
This do not seems as the correct answer for the question number 2, since after some enumeration on paper seems that there are 9 ways of arranging two rooks on that board.
Can someone provide a little of explanation or point me to a good source where I can read more about this topic? (Wiki is a little too generic)
Thanks