A Factorization Theorem for m -rook Placements Nicholas Loehr Department of Mathematics, Virginia Tech Blacksburg, VA 24061-0123 and Jeffrey Remmel Department of Mathematics, UCSD La Jolla, CA, 92093-0112 and Kenneth Barrese and Bruce Sagan Department of Mathematics, Michigan State University East Lansing, MI 48824-1027 www.math.msu.edu/˜sagan
The Factorization Theorem m -rook Placements Arbitrary Ferrers Boards Other Work
Outline The Factorization Theorem m -rook Placements Arbitrary Ferrers Boards Other Work
Consider the board gotten by tiling the first quadrant with squares . . . Q = . . .
Consider the board gotten by tiling the first quadrant with squares . . . Q = . . . Let N be the non-negative integers. A partition is a weakly increasing sequence of elements of N , denoted B = ( b 1 , . . . , b n ).
Consider the board gotten by tiling the first quadrant with squares . . . Q = . . . Let N be the non-negative integers. A partition is a weakly increasing sequence of elements of N , denoted B = ( b 1 , . . . , b n ). We also let B stand for the Ferrers board obtained by taking the lowest b j squares of Q in column j for all j .
Consider the board gotten by tiling the first quadrant with squares . . . Q = . . . Let N be the non-negative integers. A partition is a weakly increasing sequence of elements of N , denoted B = ( b 1 , . . . , b n ). We also let B stand for the Ferrers board obtained by taking the lowest b j squares of Q in column j for all j . Ex. B = (1 , 3 , 3) =
For any board B , a rook placement is a subset of B having no two squares in the same row or column.
For any board B , a rook placement is a subset of B having no two squares in the same row or column. The kth rook number of B is r k ( B ) = number of placements of k rooks on B .
For any board B , a rook placement is a subset of B having no two squares in the same row or column. The kth rook number of B is r k ( B ) = number of placements of k rooks on B . Ex. B = (1 , 3) = r 0 ( B ) = 1 , r 1 ( B ) = 4 , r 2 ( B ) = 2 .
For any board B , a rook placement is a subset of B having no two squares in the same row or column. The kth rook number of B is r k ( B ) = number of placements of k rooks on B . Let x is a variable and n ∈ N . The corresponding falling factorial is x ↓ n = x ( x − 1) · · · ( x − n + 1) . Ex. B = (1 , 3) = r 0 ( B ) = 1 , r 1 ( B ) = 4 , r 2 ( B ) = 2 .
For any board B , a rook placement is a subset of B having no two squares in the same row or column. The kth rook number of B is r k ( B ) = number of placements of k rooks on B . Let x is a variable and n ∈ N . 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 , 3) = r 0 ( B ) = 1 , r 1 ( B ) = 4 , r 2 ( B ) = 2 .
For any board B , a rook placement is a subset of B having no two squares in the same row or column. The kth rook number of B is r k ( B ) = number of placements of k rooks on B . Let x is a variable and n ∈ N . 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 , 3) = r 0 ( B ) = 1 , r 1 ( B ) = 4 , r 2 ( B ) = 2 . 2 � r k ( B ) x ↓ n − k = 1 · x ↓ 2 +4 · x ↓ 1 +2 · x ↓ 0 k =0
For any board B , a rook placement is a subset of B having no two squares in the same row or column. The kth rook number of B is r k ( B ) = number of placements of k rooks on B . Let x is a variable and n ∈ N . 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 , 3) = r 0 ( B ) = 1 , r 1 ( B ) = 4 , r 2 ( B ) = 2 . 2 � r k ( B ) x ↓ n − k = 1 · x ↓ 2 +4 · x ↓ 1 +2 · x ↓ 0 = x 2 + 3 x + 2 k =0
For any board B , a rook placement is a subset of B having no two squares in the same row or column. The kth rook number of B is r k ( B ) = number of placements of k rooks on B . Let x is a variable and n ∈ N . 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 , 3) = r 0 ( B ) = 1 , r 1 ( B ) = 4 , r 2 ( B ) = 2 . 2 � r k ( B ) x ↓ n − k = 1 · x ↓ 2 +4 · x ↓ 1 +2 · x ↓ 0 = x 2 + 3 x + 2 k =0 = ( x + 1)( x + 2)
For any board B , a rook placement is a subset of B having no two squares in the same row or column. The kth rook number of B is r k ( B ) = number of placements of k rooks on B . Let x is a variable and n ∈ N . 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 , 3) = r 0 ( B ) = 1 , r 1 ( B ) = 4 , r 2 ( B ) = 2 . 2 � r k ( B ) x ↓ n − k = 1 · x ↓ 2 +4 · x ↓ 1 +2 · x ↓ 0 = x 2 + 3 x + 2 k =0 = ( x + 1)( x + 2) = ( x + b 1 )( x + b 2 − 1) .
Outline The Factorization Theorem m -rook Placements Arbitrary Ferrers Boards Other Work
Fix a positive integer m .
Fix a positive integer m . We partition Q into levels where the j th level consists of rows jm + 1 , jm + 2 , . . . , ( j + 1) m .
Fix a positive integer m . We partition Q into levels where the j th level consists of rows jm + 1 , jm + 2 , . . . , ( j + 1) m . . . Ex. If m = 2 then . � level 1 . . . � level 0
Fix a positive integer m . We partition Q into levels where the j th level consists of rows jm + 1 , jm + 2 , . . . , ( j + 1) m . . . Ex. If m = 2 then . � level 1 . . . � level 0 An m-rook placement on B ⊆ Q is a subset of B having no two squares in the same level or column.
Fix a positive integer m . We partition Q into levels where the j th level consists of rows jm + 1 , jm + 2 , . . . , ( j + 1) m . . . Ex. If m = 2 then . � level 1 . . . � level 0 An m-rook placement on B ⊆ Q is a subset of B having no two squares in the same level or column. The kth m-rook number of B : r k , m ( B ) = number of m -rook placements on B with k rooks.
Fix a positive integer m . We partition Q into levels where the j th level consists of rows jm + 1 , jm + 2 , . . . , ( j + 1) m . . . Ex. If m = 2 then . � level 1 . . . � level 0 An m-rook placement on B ⊆ Q is a subset of B having no two squares in the same level or column. The kth m-rook number of B : r k , m ( B ) = number of m -rook placements on B with k rooks. Ex. If m = k = 2 and B = (1 , 2 , 3) =
Fix a positive integer m . We partition Q into levels where the j th level consists of rows jm + 1 , jm + 2 , . . . , ( j + 1) m . . . Ex. If m = 2 then . � level 1 . . . � level 0 An m-rook placement on B ⊆ Q is a subset of B having no two squares in the same level or column. The kth m-rook number of B : r k , m ( B ) = number of m -rook placements on B with k rooks. Ex. If m = k = 2 and B = (1 , 2 , 3) = R R R ∴ r 2 , 2 = 3 : R R R
The m -rook placements are realated to C m ≀ S k where C m is the m cyclic group and S k is the k th symmetric group,
The m -rook placements are realated to C m ≀ S k where C m is the m cyclic group and S k is the k th symmetric group, e.g., k � �� � r k , m ( mk , . . . , mk )
The m -rook placements are realated to C m ≀ S k where C m is the m cyclic group and S k is the k th symmetric group, e.g., k � �� � r k , m ( mk , . . . , mk ) = ( mk )( mk − m ) · · · ( m )
The m -rook placements are realated to C m ≀ S k where C m is the m cyclic group and S k is the k th symmetric group, e.g., k � �� � mk , . . . , mk ) = ( mk )( mk − m ) · · · ( m ) = m k k ! r k , m (
The m -rook placements are realated to C m ≀ S k where C m is the m cyclic group and S k is the k th symmetric group, e.g., k � �� � mk , . . . , mk ) = ( mk )( mk − m ) · · · ( m ) = m k k ! = # C m ≀ S k . r k , m (
The m -rook placements are realated to C m ≀ S k where C m is the m cyclic group and S k is the k th symmetric group, e.g., k � �� � mk , . . . , mk ) = ( mk )( mk − m ) · · · ( m ) = m k k ! = # C m ≀ S k . r k , m ( Define the m-falling factorials by x ↓ n , m = x ( x − m )( x − 2 m ) · · · ( x − ( n − 1) m ) .
The m -rook placements are realated to C m ≀ S k where C m is the m cyclic group and S k is the k th symmetric group, e.g., k � �� � mk , . . . , mk ) = ( mk )( mk − m ) · · · ( m ) = m k k ! = # C m ≀ S k . r k , m ( Define the m-falling factorials by x ↓ n , m = x ( x − m )( x − 2 m ) · · · ( x − ( n − 1) m ) . Given an integer m , define the mod m floor function by ⌊ n ⌋ m = largest multiple of m which is less than or equal to n .
Recommend
More recommend
