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Rook Polynomials
We will be looking at the following:
Rooks and Chessboards
Stubborn Relatives Problem
Properties of Rook Polynomials
Rook Polynomials
Presented by: Ethan Lightfoot
Rooks
A rook is a piece in chess that can move an infinite number of spaces left, right, up or down on a chessboard.
A non-taking rook is any rook that can move freely in it's row or column without taking another rook.
Rook Polynomial
A rook polynomial is a place holding function that organizes the number of rooks we place on a chessboard.
Chessboards
A chessboard is any finite board of squares.
The chessboard shape depends on the situation that we are modeling.
Can you find a rook polynomial?
Properties of Rook Polynomials
Disjoint Property
If C is a chessboard made up of pairwise disjoint subboards C1, C2, C3, … ,Cn, then r(C,x) = r(C1,x)*r(C2,x)*r(C3,x)* *… *r(Cn,x).
Disjoint Subboard Property
A subboard is any board whose squares are contained by a larger chess board.
Disjoint Subboard Property
Subboards are considered disjoint if and only if the subboards do not share squares which lie in the same row or column.
Proof: Fermat Method
Column/Row Permutation Property
The rows or columns of any chessboard may be permuted and the integrity of the chessboard will still hold.
This property holds as long as the conditions of the original chessboard are still satisfied.
Column/Row Permutation Property
Stubborn Relatives
Conditions
For the wedding reception, almost all of the family and friends have been placed at tables except for four relatives.
Dr. Daly
Let Ri, for iϵ[4], denote the relatives needing to be placed and let Tj, for jϵ[5], represent the five tables at which the relatives may be placed. Because of family differences,
These relatives have not been placed yet since they seem to have problems with some of the other family members at some of the tables.
1)R1 may not be placed at T1 or T2.
2)R2 may not be placed at T2.
3)R3 may not be placed at T3 or T4.
There are only five seats available for these relatives and each of these seats is at a different table.
4)R4 may not be placed at T4 or T5.
We want to know, in how many ways can all four relatives be placed at a table such that all conditions are satisfied.
Jennifer Connelly
Ethan
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