Factoring rook polynomials Bruce Sagan Department of Mathematics, - PowerPoint PPT Presentation

Factoring rook polynomials Bruce Sagan Department of Mathematics, Michigan State University East Lansing, MI 48824-1027 www.math.msu.edu/˜sagan March 14, 2014

Basics The FactorizationTheorem An application Exercises and References

Outline Basics The FactorizationTheorem An application Exercises and References

Consider tiling the first quadrant of the plane with unit squares: . . . . . . Q =

Consider tiling the first quadrant of the plane with unit squares: . . . . . . Q = Let ( c , d ) be the square in column c and row d .

Consider tiling the first quadrant of the plane with unit squares: . . . (3 , 4) . . . Q = Let ( c , d ) be the square in column c and row d .

Consider tiling the first quadrant of the plane with unit squares: . . . (3 , 4) . . . Q = Let ( c , d ) be the square in column c and row d . A board is a finite set of squares B ⊆ Q .

Consider tiling the first quadrant of the plane with unit squares: . . . (3 , 4) . . . Q = Let ( c , d ) be the square in column c and row d . A board is a finite set of squares B ⊆ Q . Ex. Let B n be the n × n chess board. For example, B 3 =

Consider tiling the first quadrant of the plane with unit squares: . . . (3 , 4) . . . Q = Let ( c , d ) be the square in column c and row d . A board is a finite set of squares B ⊆ Q . Ex. Let B n be the n × n chess board. For example, B 3 = A placement P of rooks on B is attacking if there is a pair of rooks in the same row or column. Otherwise it is nonattacking .

Consider tiling the first quadrant of the plane with unit squares: . . . (3 , 4) . . . Q = Let ( c , d ) be the square in column c and row d . A board is a finite set of squares B ⊆ Q . Ex. Let B n be the n × n chess board. For example, attacking: R P = R R B 3 = A placement P of rooks on B is attacking if there is a pair of rooks in the same row or column. Otherwise it is nonattacking .

Consider tiling the first quadrant of the plane with unit squares: . . . (3 , 4) . . . Q = Let ( c , d ) be the square in column c and row d . A board is a finite set of squares B ⊆ Q . Ex. Let B n be the n × n chess board. For example, attacking: nonattacking: R R P = R R R B 3 = P = R A placement P of rooks on B is attacking if there is a pair of rooks in the same row or column. Otherwise it is nonattacking .

Define the rook numbers of B to be r k ( B ) = number of ways of placing k nonattacking rooks on B .

Define the rook numbers of B to be r k ( B ) = number of ways of placing k nonattacking rooks on B . For any board B we have r 0 ( B ) = 1

Define the rook numbers of B to be r k ( B ) = number of ways of placing k nonattacking rooks on B . For any board B we have r 0 ( B ) = 1 and r 1 ( B ) = | B | (cardinality).

Define the rook numbers of B to be r k ( B ) = number of ways of placing k nonattacking rooks on B . For any board B we have r 0 ( B ) = 1 and r 1 ( B ) = | B | (cardinality). Ex. We have r n ( B n )

Define the rook numbers of B to be r k ( B ) = number of ways of placing k nonattacking rooks on B . For any board B we have r 0 ( B ) = 1 and r 1 ( B ) = | B | (cardinality). Ex. We have r n ( B n ) = (# of ways to place a rook in column 1) · (# of ways to then place a rook in column 2) · · ·

Define the rook numbers of B to be r k ( B ) = number of ways of placing k nonattacking rooks on B . For any board B we have r 0 ( B ) = 1 and r 1 ( B ) = | B | (cardinality). Ex. We have r n ( B n ) = (# of ways to place a rook in column 1) · (# of ways to then place a rook in column 2) · · · = n · ( n − 1) · · ·

Define the rook numbers of B to be r k ( B ) = number of ways of placing k nonattacking rooks on B . For any board B we have r 0 ( B ) = 1 and r 1 ( B ) = | B | (cardinality). Ex. We have r n ( B n ) = (# of ways to place a rook in column 1) · (# of ways to then place a rook in column 2) · · · = n · ( n − 1) · · · = n !

Define the rook numbers of B to be r k ( B ) = number of ways of placing k nonattacking rooks on B . For any board B we have r 0 ( B ) = 1 and r 1 ( B ) = | B | (cardinality). Ex. We have r n ( B n ) = (# of ways to place a rook in column 1) · (# of ways to then place a rook in column 2) · · · = n · ( n − 1) · · · = n ! There is a bijection between placements P counted by r n ( B n ) and permutations π in the symmetric group S n where ( c , d ) ∈ P if and only if π ( c ) = d .

Define the rook numbers of B to be r k ( B ) = number of ways of placing k nonattacking rooks on B . For any board B we have r 0 ( B ) = 1 and r 1 ( B ) = | B | (cardinality). Ex. We have r n ( B n ) = (# of ways to place a rook in column 1) · (# of ways to then place a rook in column 2) · · · = n · ( n − 1) · · · = n ! There is a bijection between placements P counted by r n ( B n ) and permutations π in the symmetric group S n where ( c , d ) ∈ P if and only if π ( c ) = d . Ex. Let D n = B n − { (1 , 1) , (2 , 2) , . . . , ( n , n ) } .

Define the rook numbers of B to be r k ( B ) = number of ways of placing k nonattacking rooks on B . For any board B we have r 0 ( B ) = 1 and r 1 ( B ) = | B | (cardinality). Ex. We have r n ( B n ) = (# of ways to place a rook in column 1) · (# of ways to then place a rook in column 2) · · · = n · ( n − 1) · · · = n ! There is a bijection between placements P counted by r n ( B n ) and permutations π in the symmetric group S n where ( c , d ) ∈ P if and only if π ( c ) = d . Ex. Let D n = B n − { (1 , 1) , (2 , 2) , . . . , ( n , n ) } . Then r n ( D n )

Define the rook numbers of B to be r k ( B ) = number of ways of placing k nonattacking rooks on B . For any board B we have r 0 ( B ) = 1 and r 1 ( B ) = | B | (cardinality). Ex. We have r n ( B n ) = (# of ways to place a rook in column 1) · (# of ways to then place a rook in column 2) · · · = n · ( n − 1) · · · = n ! There is a bijection between placements P counted by r n ( B n ) and permutations π in the symmetric group S n where ( c , d ) ∈ P if and only if π ( c ) = d . Ex. Let D n = B n − { (1 , 1) , (2 , 2) , . . . , ( n , n ) } . Then r n ( D n ) = # of permutations π ∈ S n with π ( c ) � = c for all c

Define the rook numbers of B to be r k ( B ) = number of ways of placing k nonattacking rooks on B . For any board B we have r 0 ( B ) = 1 and r 1 ( B ) = | B | (cardinality). Ex. We have r n ( B n ) = (# of ways to place a rook in column 1) · (# of ways to then place a rook in column 2) · · · = n · ( n − 1) · · · = n ! There is a bijection between placements P counted by r n ( B n ) and permutations π in the symmetric group S n where ( c , d ) ∈ P if and only if π ( c ) = d . Ex. Let D n = B n − { (1 , 1) , (2 , 2) , . . . , ( n , n ) } . Then r n ( D n ) = # of permutations π ∈ S n with π ( c ) � = c for all c = the n th derangement number.

Outline Basics The FactorizationTheorem An application Exercises and References

A partition is a weakly increasing sequence ( b 1 , . . . , b n ) of nonnegative integers.

A partition is a weakly increasing sequence ( b 1 , . . . , b n ) of nonnegative integers. A Ferrers board is B = ( b 1 , . . . , b n ) consisting of the lowest b j squares in column j of Q for all j .

A partition is a weakly increasing sequence ( b 1 , . . . , b n ) of nonnegative integers. A Ferrers board is B = ( b 1 , . . . , b n ) consisting of the lowest b j squares in column j of Q for all j . Ex. B = (1 , 1 , 3) =

A partition is a weakly increasing sequence ( b 1 , . . . , b n ) of nonnegative integers. A Ferrers board is B = ( b 1 , . . . , b n ) consisting of the lowest b j squares in column j of Q for all j . If x is a variable and n ≥ 0 then the corresponding falling factorial is x ↓ n = x ( x − 1) · · · ( x − n + 1) . Ex. B = (1 , 1 , 3) =

A partition is a weakly increasing sequence ( b 1 , . . . , b n ) of nonnegative integers. A Ferrers board is B = ( b 1 , . . . , b n ) consisting of the lowest b j squares in column j of Q for all j . If x is a variable and n ≥ 0 then the corresponding falling factorial is x ↓ n = x ( x − 1) · · · ( x − n + 1) . Theorem (Factorization Theorem: Goldman-Joichi-White) For any Ferrers board B = ( b 1 , . . . , b n ) we have n n � � r k ( B ) x ↓ n − k = ( x + b j − j + 1) . k =0 j =1 Ex. B = (1 , 1 , 3) =

A partition is a weakly increasing sequence ( b 1 , . . . , b n ) of nonnegative integers. A Ferrers board is B = ( b 1 , . . . , b n ) consisting of the lowest b j squares in column j of Q for all j . If x is a variable and n ≥ 0 then the corresponding falling factorial is x ↓ n = x ( x − 1) · · · ( x − n + 1) . Theorem (Factorization Theorem: Goldman-Joichi-White) For any Ferrers board B = ( b 1 , . . . , b n ) we have n n � � r k ( B ) x ↓ n − k = ( x + b j − j + 1) . k =0 j =1 Ex. B = (1 , 1 , 3) = r 0 ( B ) = 1 , r 1 ( B ) = 5 , r 2 ( B ) = 4 , r 3 ( B ) = 0.