Alternating Sign Matrices and Latin Squares
Cian O’Brien Rachel Quinlan and Kevin Jennings
National University of Ireland, Galway [email protected]
Postgraduate Modelling Group, NUI Galway October 4th, 2019
Cian O’Brien (NUIG) ASBG-Colourings October 4th, 2019 1 / 9 0 1 0 0 1 −1 1 0 0 1 −1 1 0 0 1 0 They are a generalisation of the permutation matrices.
+ 0 0 0 0 0 0 + 0 + 0 0 0 0 + 0
Alternating Sign Matrices
An Alternating Sign Matrix (ASM) is (0, 1, −1)-matrix for which all row and column sums are 1, and the non-zero elements in each row and column alternate in sign.
Cian O’Brien (NUIG) ASBG-Colourings October 4th, 2019 2 / 9 They are a generalisation of the permutation matrices.
+ 0 0 0 0 0 0 + 0 + 0 0 0 0 + 0
Alternating Sign Matrices
An Alternating Sign Matrix (ASM) is (0, 1, −1)-matrix for which all row and column sums are 1, and the non-zero elements in each row and column alternate in sign.
0 1 0 0 1 −1 1 0 0 1 −1 1 0 0 1 0
Cian O’Brien (NUIG) ASBG-Colourings October 4th, 2019 2 / 9 They are a generalisation of the permutation matrices.
+ 0 0 0 0 0 0 + 0 + 0 0 0 0 + 0
Alternating Sign Matrices
An Alternating Sign Matrix (ASM) is (0, 1, −1)-matrix for which all row and column sums are 1, and the non-zero elements in each row and column alternate in sign.
0 + 0 0 + − + 0 0 + − + 0 0 + 0
Cian O’Brien (NUIG) ASBG-Colourings October 4th, 2019 2 / 9 + 0 0 0 0 0 0 + 0 + 0 0 0 0 + 0
Alternating Sign Matrices
An Alternating Sign Matrix (ASM) is (0, 1, −1)-matrix for which all row and column sums are 1, and the non-zero elements in each row and column alternate in sign.
0 + 0 0 + − + 0 0 + − + 0 0 + 0 They are a generalisation of the permutation matrices.
Cian O’Brien (NUIG) ASBG-Colourings October 4th, 2019 2 / 9 Alternating Sign Matrices
An Alternating Sign Matrix (ASM) is (0, 1, −1)-matrix for which all row and column sums are 1, and the non-zero elements in each row and column alternate in sign.
0 + 0 0 + − + 0 0 + − + 0 0 + 0 They are a generalisation of the permutation matrices.
+ 0 0 0 0 0 0 + 0 + 0 0 0 0 + 0
Cian O’Brien (NUIG) ASBG-Colourings October 4th, 2019 2 / 9 + 0 0 0 + 0 0 0 + 0 + 0 % + − + % 0 + 0 0 0 + 0 + 0 + 0 0
Alternating Sign Hypermatrices
An Alternating Sign Hypermatrix (ASHM) is a hypermatrix (n × n × n array) for which every plane is an ASM.
Cian O’Brien (NUIG) ASBG-Colourings October 4th, 2019 3 / 9 + 0 0 0 + 0 0 0 + 0 + 0 % + − + % 0 + 0 0 0 + 0 + 0 + 0 0
Alternating Sign Hypermatrices
An Alternating Sign Hypermatrix (ASHM) is a hypermatrix (n × n × n array) for which every plane is an ASM.
Cian O’Brien (NUIG) ASBG-Colourings October 4th, 2019 3 / 9 Alternating Sign Hypermatrices
An Alternating Sign Hypermatrix (ASHM) is a hypermatrix (n × n × n array) for which every plane is an ASM.
+ 0 0 0 + 0 0 0 + 0 + 0 % + − + % 0 + 0 0 0 + 0 + 0 + 0 0
Cian O’Brien (NUIG) ASBG-Colourings October 4th, 2019 3 / 9 0 0 + 0 0 0 + − + 0 + − + − + 0 + − + 0 0 0 + 0 0
Non-Zero Entry Bounds for ASMs
The number of non-zero entries in the rows/columns of an ASM is bounded above by (1, 3, 5,..., 5, 3, 1).
Cian O’Brien (NUIG) ASBG-Colourings October 4th, 2019 4 / 9 Non-Zero Entry Bounds for ASMs
The number of non-zero entries in the rows/columns of an ASM is bounded above by (1, 3, 5,..., 5, 3, 1).
0 0 + 0 0 0 + − + 0 + − + − + 0 + − + 0 0 0 + 0 0
Cian O’Brien (NUIG) ASBG-Colourings October 4th, 2019 4 / 9 + 0 0 0 0 ! 0 + 0 0 0 ! 0 0 + 0 0 ! 0 0 0 + 0 ! 0 0 0 0 +! 0 + 0 0 0 + − + 0 0 0 + − + 0 0 0 + − + 0 0 0 + 0 0 0 + 0 0 % 0 + − + 0 % + − + − + % 0 + − + 0 % 0 0 + 0 0 0 0 0 + 0 0 0 + − + 0 + − + 0 + − + 0 0 0 + 0 0 0 0 0 0 0 + 0 0 0 + 0 0 0 + 0 0 0 + 0 0 0 + 0 0 0 0
Non-Zero Entry Bounds for ASHMs
The number of non-zero entries in the planes of an ASHM is bounded above by 1 1 ··· 1 1 1 3 ··· 3 1 ...... . . . . . 1 3 ··· 3 1 1 1 ··· 1 1
Cian O’Brien (NUIG) ASBG-Colourings October 4th, 2019 5 / 9 Non-Zero Entry Bounds for ASHMs
The number of non-zero entries in the planes of an ASHM is bounded above by 1 1 ··· 1 1 1 3 ··· 3 1 ...... . . . . . 1 3 ··· 3 1 1 1 ··· 1 1
+ 0 0 0 0 ! 0 + 0 0 0 ! 0 0 + 0 0 ! 0 0 0 + 0 ! 0 0 0 0 +! 0 + 0 0 0 + − + 0 0 0 + − + 0 0 0 + − + 0 0 0 + 0 0 0 + 0 0 % 0 + − + 0 % + − + − + % 0 + − + 0 % 0 0 + 0 0 0 0 0 + 0 0 0 + − + 0 + − + 0 + − + 0 0 0 + 0 0 0 0 0 0 0 + 0 0 0 + 0 0 0 + 0 0 0 + 0 0 0 + 0 0 0 0
Cian O’Brien (NUIG) ASBG-Colourings October 4th, 2019 5 / 9 1 2 3 2 3 1 3 1 2 Each n × n latin square can be decomposed uniquely into a sum of scalar multiples of mutually orthogonal permutation matrices. Therefore each latin square corresponds uniquely to a permutation hypermatrix. For a permutation hypermatrix M, define L(M) to be Pn L(M)i,j,k = k=1 k × Mi,j,k .
+ 0 0 0 + 0 0 0 + 0 0 + % + 0 0 % 0 + 0 0 + 0 0 0 + + 0 0
Latin Squares
An n × n Latin Square is an n × n array of n symbols such that each symbol occurs exactly once in each row and column.
Cian O’Brien (NUIG) ASBG-Colourings October 4th, 2019 6 / 9 Each n × n latin square can be decomposed uniquely into a sum of scalar multiples of mutually orthogonal permutation matrices. Therefore each latin square corresponds uniquely to a permutation hypermatrix. For a permutation hypermatrix M, define L(M) to be Pn L(M)i,j,k = k=1 k × Mi,j,k .
+ 0 0 0 + 0 0 0 + 0 0 + % + 0 0 % 0 + 0 0 + 0 0 0 + + 0 0
Latin Squares
An n × n Latin Square is an n × n array of n symbols such that each symbol occurs exactly once in each row and column.
1 2 3 2 3 1 3 1 2
Cian O’Brien (NUIG) ASBG-Colourings October 4th, 2019 6 / 9 Therefore each latin square corresponds uniquely to a permutation hypermatrix. For a permutation hypermatrix M, define L(M) to be Pn L(M)i,j,k = k=1 k × Mi,j,k .
+ 0 0 0 + 0 0 0 + 0 0 + % + 0 0 % 0 + 0 0 + 0 0 0 + + 0 0
Latin Squares
An n × n Latin Square is an n × n array of n symbols such that each symbol occurs exactly once in each row and column.
1 2 3 2 3 1 3 1 2 Each n × n latin square can be decomposed uniquely into a sum of scalar multiples of mutually orthogonal permutation matrices.
Cian O’Brien (NUIG) ASBG-Colourings October 4th, 2019 6 / 9 Therefore each latin square corresponds uniquely to a permutation hypermatrix. For a permutation hypermatrix M, define L(M) to be Pn L(M)i,j,k = k=1 k × Mi,j,k .
+ 0 0 0 + 0 0 0 + 0 0 + % + 0 0 % 0 + 0 0 + 0 0 0 + + 0 0
Latin Squares
An n × n Latin Square is an n × n array of n symbols such that each symbol occurs exactly once in each row and column.
1 2 3 1 0 0 0 1 0 0 0 1 2 3 1 = 0 0 1 + 2 1 0 0 + 3 0 1 0 3 1 2 0 1 0 0 0 1 1 0 0 Each n × n latin square can be decomposed uniquely into a sum of scalar multiples of mutually orthogonal permutation matrices.
Cian O’Brien (NUIG) ASBG-Colourings October 4th, 2019 6 / 9 + 0 0 0 + 0 0 0 + 0 0 + % + 0 0 % 0 + 0 0 + 0 0 0 + + 0 0
Latin Squares
An n × n Latin Square is an n × n array of n symbols such that each symbol occurs exactly once in each row and column.
1 2 3 1 0 0 0 1 0 0 0 1 2 3 1 = 0 0 1 + 2 1 0 0 + 3 0 1 0 3 1 2 0 1 0 0 0 1 1 0 0 Each n × n latin square can be decomposed uniquely into a sum of scalar multiples of mutually orthogonal permutation matrices. Therefore each latin square corresponds uniquely to a permutation hypermatrix. For a permutation hypermatrix M, define L(M) to be Pn L(M)i,j,k = k=1 k × Mi,j,k .
Cian O’Brien (NUIG) ASBG-Colourings October 4th, 2019 6 / 9 Latin Squares
An n × n Latin Square is an n × n array of n symbols such that each symbol occurs exactly once in each row and column.
1 2 3 1 0 0 0 1 0 0 0 1 2 3 1 = 0 0 1 + 2 1 0 0 + 3 0 1 0 3 1 2 0 1 0 0 0 1 1 0 0 Each n × n latin square can be decomposed uniquely into a sum of scalar multiples of mutually orthogonal permutation matrices. Therefore each latin square corresponds uniquely to a permutation hypermatrix. For a permutation hypermatrix M, define L(M) to be Pn L(M)i,j,k = k=1 k × Mi,j,k .
+ 0 0 0 + 0 0 0 + 0 0 + % + 0 0 % 0 + 0 0 + 0 0 0 + + 0 0
Cian O’Brien (NUIG) ASBG-Colourings October 4th, 2019 6 / 9 + 0 0 0 + 0 0 0 + A = 0 + 0 % + − + % 0 + 0 0 0 + 0 + 0 + 0 0
1 2 3 L(A) = 2 2 2 3 2 1
ASHM-Latin Squares
An n × n ASHM-Latin Square is an n × n matrix L(A) such that Pn L(A)i,j,k = k=1 k × Ai,j,k for some n × n × n alternating sign hypermatrix A
Cian O’Brien (NUIG) ASBG-Colourings October 4th, 2019 7 / 9 1 2 3 L(A) = 2 2 2 3 2 1
ASHM-Latin Squares
An n × n ASHM-Latin Square is an n × n matrix L(A) such that Pn L(A)i,j,k = k=1 k × Ai,j,k for some n × n × n alternating sign hypermatrix A
+ 0 0 0 + 0 0 0 + A = 0 + 0 % + − + % 0 + 0 0 0 + 0 + 0 + 0 0
Cian O’Brien (NUIG) ASBG-Colourings October 4th, 2019 7 / 9 ASHM-Latin Squares
An n × n ASHM-Latin Square is an n × n matrix L(A) such that Pn L(A)i,j,k = k=1 k × Ai,j,k for some n × n × n alternating sign hypermatrix A
+ 0 0 0 + 0 0 0 + A = 0 + 0 % + − + % 0 + 0 0 0 + 0 + 0 + 0 0
1 2 3 L(A) = 2 2 2 3 2 1
Cian O’Brien (NUIG) ASBG-Colourings October 4th, 2019 7 / 9 The outer planes of the ASHM can each only contain one entry.
Each row and column sum of the ASHM is 1, and row and column of and ASHM-Latin Square is therefore 1(1) + 2(1) + 3(1) + ··· + n(1).
An entry of an ASHM-Latin Square is calculated by the sum of integers between 1 and n which increase in magnitude but alternate in sign.
The outer rows and columns contain each symbol exactly once.
n(n+1) Each row and column sums to 2 .
Basic Facts about ASHM-Latin Squares
All entries of an n × n ASHM-Latin Square are between 1 and n.
Cian O’Brien (NUIG) ASBG-Colourings October 4th, 2019 8 / 9 The outer planes of the ASHM can each only contain one entry.
Each row and column sum of the ASHM is 1, and row and column of and ASHM-Latin Square is therefore 1(1) + 2(1) + 3(1) + ··· + n(1).
The outer rows and columns contain each symbol exactly once.
n(n+1) Each row and column sums to 2 .
Basic Facts about ASHM-Latin Squares
All entries of an n × n ASHM-Latin Square are between 1 and n. An entry of an ASHM-Latin Square is calculated by the sum of integers between 1 and n which increase in magnitude but alternate in sign.
Cian O’Brien (NUIG) ASBG-Colourings October 4th, 2019 8 / 9 Each row and column sum of the ASHM is 1, and row and column of and ASHM-Latin Square is therefore 1(1) + 2(1) + 3(1) + ··· + n(1).
The outer planes of the ASHM can each only contain one entry.
n(n+1) Each row and column sums to 2 .
Basic Facts about ASHM-Latin Squares
All entries of an n × n ASHM-Latin Square are between 1 and n. An entry of an ASHM-Latin Square is calculated by the sum of integers between 1 and n which increase in magnitude but alternate in sign.
The outer rows and columns contain each symbol exactly once.
Cian O’Brien (NUIG) ASBG-Colourings October 4th, 2019 8 / 9 Each row and column sum of the ASHM is 1, and row and column of and ASHM-Latin Square is therefore 1(1) + 2(1) + 3(1) + ··· + n(1).
n(n+1) Each row and column sums to 2 .
Basic Facts about ASHM-Latin Squares
All entries of an n × n ASHM-Latin Square are between 1 and n. An entry of an ASHM-Latin Square is calculated by the sum of integers between 1 and n which increase in magnitude but alternate in sign.
The outer rows and columns contain each symbol exactly once. The outer planes of the ASHM can each only contain one entry.
Cian O’Brien (NUIG) ASBG-Colourings October 4th, 2019 8 / 9 Each row and column sum of the ASHM is 1, and row and column of and ASHM-Latin Square is therefore 1(1) + 2(1) + 3(1) + ··· + n(1).
Basic Facts about ASHM-Latin Squares
All entries of an n × n ASHM-Latin Square are between 1 and n. An entry of an ASHM-Latin Square is calculated by the sum of integers between 1 and n which increase in magnitude but alternate in sign.
The outer rows and columns contain each symbol exactly once. The outer planes of the ASHM can each only contain one entry.
n(n+1) Each row and column sums to 2 .
Cian O’Brien (NUIG) ASBG-Colourings October 4th, 2019 8 / 9 Basic Facts about ASHM-Latin Squares
All entries of an n × n ASHM-Latin Square are between 1 and n. An entry of an ASHM-Latin Square is calculated by the sum of integers between 1 and n which increase in magnitude but alternate in sign.
The outer rows and columns contain each symbol exactly once. The outer planes of the ASHM can each only contain one entry.
n(n+1) Each row and column sums to 2 . Each row and column sum of the ASHM is 1, and row and column of and ASHM-Latin Square is therefore 1(1) + 2(1) + 3(1) + ··· + n(1).
Cian O’Brien (NUIG) ASBG-Colourings October 4th, 2019 8 / 9 Each row and column of an ASHM-Latin Square is majorized by zn = (n, n − 1,..., 2, 1).
Current highest found is 2n, with achievable example for all odd n ≥ 5
1 237564 2 233675 3325357 4342636 6 734233 7 5 63322 5 6 7 4321
If L(A) is a latin square, then A is a permutation matrix. A full characterisation of ASHM-Latin Squares.
What is the maximum number of times a symbol can occur in a ASHM-Latin Square?
Interesting Open Problems/Questions
Does each ASHM-Latin Square correspond to exactly one ASHM?
Cian O’Brien (NUIG) ASBG-Colourings October 4th, 2019 9 / 9 Each row and column of an ASHM-Latin Square is majorized by zn = (n, n − 1,..., 2, 1).
Current highest found is 2n, with achievable example for all odd n ≥ 5
1 237564 2 233675 3325357 4342636 6 734233 7 5 63322 5 6 7 4321
A full characterisation of ASHM-Latin Squares.
What is the maximum number of times a symbol can occur in a ASHM-Latin Square?
Interesting Open Problems/Questions
Does each ASHM-Latin Square correspond to exactly one ASHM? If L(A) is a latin square, then A is a permutation matrix.
Cian O’Brien (NUIG) ASBG-Colourings October 4th, 2019 9 / 9 Current highest found is 2n, with achievable example for all odd n ≥ 5
1 237564 2 233675 3325357 4342636 6 734233 7 5 63322 5 6 7 4321
Each row and column of an ASHM-Latin Square is majorized by zn = (n, n − 1,..., 2, 1). What is the maximum number of times a symbol can occur in a ASHM-Latin Square?
Interesting Open Problems/Questions
Does each ASHM-Latin Square correspond to exactly one ASHM? If L(A) is a latin square, then A is a permutation matrix. A full characterisation of ASHM-Latin Squares.
Cian O’Brien (NUIG) ASBG-Colourings October 4th, 2019 9 / 9 Current highest found is 2n, with achievable example for all odd n ≥ 5
1 237564 2 233675 3325357 4342636 6 734233 7 5 63322 5 6 7 4321
What is the maximum number of times a symbol can occur in a ASHM-Latin Square?
Interesting Open Problems/Questions
Does each ASHM-Latin Square correspond to exactly one ASHM? If L(A) is a latin square, then A is a permutation matrix. A full characterisation of ASHM-Latin Squares. Each row and column of an ASHM-Latin Square is majorized by zn = (n, n − 1,..., 2, 1).
Cian O’Brien (NUIG) ASBG-Colourings October 4th, 2019 9 / 9 Current highest found is 2n, with achievable example for all odd n ≥ 5
1 237564 2 233675 3325357 4342636 6 734233 7 5 63322 5 6 7 4321
Interesting Open Problems/Questions
Does each ASHM-Latin Square correspond to exactly one ASHM? If L(A) is a latin square, then A is a permutation matrix. A full characterisation of ASHM-Latin Squares. Each row and column of an ASHM-Latin Square is majorized by zn = (n, n − 1,..., 2, 1). What is the maximum number of times a symbol can occur in a ASHM-Latin Square?
Cian O’Brien (NUIG) ASBG-Colourings October 4th, 2019 9 / 9 Interesting Open Problems/Questions
Does each ASHM-Latin Square correspond to exactly one ASHM? If L(A) is a latin square, then A is a permutation matrix. A full characterisation of ASHM-Latin Squares. Each row and column of an ASHM-Latin Square is majorized by zn = (n, n − 1,..., 2, 1). What is the maximum number of times a symbol can occur in a ASHM-Latin Square? Current highest found is 2n, with achievable example for all odd n ≥ 5
1 237564 2 233675 3325357 4342636 6 734233 7 5 63322 5 6 7 4321
Cian O’Brien (NUIG) ASBG-Colourings October 4th, 2019 9 / 9 Richard A. Brualdi, Geir Dahl, Alternating Sign Matrices and Hypermatrices, and a Generalization of Latin Squares. arXiv:1704.07752, 2017.
Cian O’Brien (NUIG) ASBG-Colourings October 4th, 2019 9 / 9