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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

fjanisze
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    The numbers you seek are the coefficients of the rook polynomial, not its values when you put in $1$, $2$, $3$. – Angina Seng Apr 20 '17 at 06:55
  • Perhas you would be interested in my answer to this question. – N. Shales Apr 20 '17 at 09:49
  • If you read that you will see how rook polynomials can be used to solve a problem. However, it seems to be your goal simply to calculate the coefficients of the rook polynomial. In this case I would use the complement board to calculate those coefficients but it isn't an introductory type question. Also you have somehow already calculated those coefficients (i.e. the answers to 1,2 and 3) so what's the point of having the rook polynomial at all? How did you calculate those coefficients btw? – N. Shales Apr 20 '17 at 10:29
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    Thanks all for you comments, @N. Shales I will have a look at your post for sure. I found those coefficient by enumerating the possible configuration on paper. I was sure that the generating function is needed to perform some calculation for this exercise, not that it's only needed to extrapolate the coefficients, the question statement get me confuse on this. – fjanisze Apr 21 '17 at 06:29

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