Richard Brualdi - University of Regina

Graphs, Designs and Algebraic Combinatorics, 18–21 July 2011
Alternating
Sign Matrices
and Their
Patterns
Richard A.
Brualdi
Alternating Sign Matrices and Their Patterns
Alternating
Sign Matrices
(ASMs)
Patterns of
ASMs
Symmetric
ASMs
Additional
Properties of
ASMs
Richard A. Brualdi
University of Wisconsin-Madison, USA
Joint work with K.P. Kiernan, S.A. Meyer, M.W. Schroeder
Graphs, Designs and Algebraic Combinatorics
18–21 July 2011
University of Regina

Graphs, Designs and Algebraic Combinatorics, 18–21 July 2011
Alternating
Sign Matrices
and Their
Patterns
Richard A.
Brualdi
1 Alternating Sign Matrices (ASMs)
Alternating
Sign Matrices
(ASMs)
Patterns of
ASMs
Symmetric
ASMs
Additional
Properties of
ASMs
2 Patterns of ASMs
3 Symmetric ASMs
4 Additional Properties of ASMs

Graphs, Designs and Algebraic Combinatorics, 18–21 July 2011
What is an ASM?
Alternating
Sign Matrices
and Their
Patterns
Richard A.
Brualdi
Alternating
Sign Matrices
(ASMs)
Patterns of
ASMs
Symmetric
ASMs
Additional
Properties of
ASMs
n × n (0, 1, −1)-matrix such that in each row and column the
1s and −1s alternate beginning and ending with 1.
Examples:
Permutation matrices.
⎡
⎤ ⎡
⎤
0 1 0 0 + 0
⎣ 1 −1 1 ⎦ −→ ⎣ + − + ⎦.
0 1 0 0 + 0
⎡
⎤
0 0 + 0 0
0 + − + 0
⎢ + − + − +
⎥
⎣ 0 + − + 0 ⎦
0 0 + 0 0

Graphs, Designs and Algebraic Combinatorics, 18–21 July 2011
What is an ASM?
Alternating
Sign Matrices
and Their
Patterns
Richard A.
Brualdi
Alternating
Sign Matrices
(ASMs)
Patterns of
ASMs
Symmetric
ASMs
Additional
Properties of
ASMs
n × n (0, 1, −1)-matrix such that in each row and column the
1s and −1s alternate beginning and ending with 1.
Examples:
Permutation matrices.
⎡
⎤ ⎡
⎤
0 1 0 0 + 0
⎣ 1 −1 1 ⎦ −→ ⎣ + − + ⎦.
0 1 0 0 + 0
⎡
⎤
0 0 + 0 0
0 + − + 0
⎢ + − + − +
⎥
⎣ 0 + − + 0 ⎦
0 0 + 0 0

Graphs, Designs and Algebraic Combinatorics, 18–21 July 2011
What is an ASM?
Alternating
Sign Matrices
and Their
Patterns
Richard A.
Brualdi
Alternating
Sign Matrices
(ASMs)
Patterns of
ASMs
Symmetric
ASMs
Additional
Properties of
ASMs
n × n (0, 1, −1)-matrix such that in each row and column the
1s and −1s alternate beginning and ending with 1.
Examples:
Permutation matrices.
⎡
⎤ ⎡
⎤
0 1 0 0 + 0
⎣ 1 −1 1 ⎦ −→ ⎣ + − + ⎦.
0 1 0 0 + 0
⎡
⎤
0 0 + 0 0
0 + − + 0
⎢ + − + − +
⎥
⎣ 0 + − + 0 ⎦
0 0 + 0 0

Graphs, Designs and Algebraic Combinatorics, 18–21 July 2011
What is an ASM?
Alternating
Sign Matrices
and Their
Patterns
Richard A.
Brualdi
Alternating
Sign Matrices
(ASMs)
Patterns of
ASMs
Symmetric
ASMs
Additional
Properties of
ASMs
n × n (0, 1, −1)-matrix such that in each row and column the
1s and −1s alternate beginning and ending with 1.
Examples:
Permutation matrices.
⎡
⎤ ⎡
⎤
0 1 0 0 + 0
⎣ 1 −1 1 ⎦ −→ ⎣ + − + ⎦.
0 1 0 0 + 0
⎡
⎤
0 0 + 0 0
0 + − + 0
⎢ + − + − +
⎥
⎣ 0 + − + 0 ⎦
0 0 + 0 0

Graphs, Designs and Algebraic Combinatorics, 18–21 July 2011
What is an ASM?
Alternating
Sign Matrices
and Their
Patterns
Richard A.
Brualdi
Alternating
Sign Matrices
(ASMs)
Patterns of
ASMs
Symmetric
ASMs
Additional
Properties of
ASMs
n × n (0, 1, −1)-matrix such that in each row and column the
1s and −1s alternate beginning and ending with 1.
Examples:
Permutation matrices.
⎡
⎤ ⎡
⎤
0 1 0 0 + 0
⎣ 1 −1 1 ⎦ −→ ⎣ + − + ⎦.
0 1 0 0 + 0
⎡
⎤
0 0 + 0 0
0 + − + 0
⎢ + − + − +
⎥
⎣ 0 + − + 0 ⎦
0 0 + 0 0

Graphs, Designs and Algebraic Combinatorics, 18–21 July 2011
Another Example of an ASM
Alternating
Sign Matrices
and Their
Patterns
Richard A.
Brualdi
Alternating
Sign Matrices
(ASMs)
Patterns of
ASMs
Symmetric
ASMs
Additional
Properties of
ASMs
⎡
⎢
⎣
+
+ − +
+
+ − +
+ − + − +
+
+
+
⎤
.
⎥
⎦

Basic Properties of ASMs
Alternating
Sign Matrices
and Their
Patterns
Richard A.
Brualdi
Alternating
Sign Matrices
(ASMs)
Patterns of
ASMs
Symmetric
ASMs
Additional
Properties of
ASMs
The first and last rows and columns in an ASM always
contain exactly one + and no −.
The number of nonzeros in each row and column is odd.
all row and column sums equal 1.
(number of +’s) = n + (number of −’s).
The partial row (column) sums starting from the first or
last entry in a row (column) equal 0 or 1.
Invariant under the dihedral group of order 8 (symmetries
of a square).
⎡ ⎤
+
⎢ ⎢⎢⎢⎢⎢⎣
+ − +
+ − + − +
+ − + − +
⎥
+ − + ⎦
+
Graphs, Designs and Algebraic Combinatorics, 18–21 July 2011

Basic Properties of ASMs
Alternating
Sign Matrices
and Their
Patterns
Richard A.
Brualdi
Alternating
Sign Matrices
(ASMs)
Patterns of
ASMs
Symmetric
ASMs
Additional
Properties of
ASMs
The first and last rows and columns in an ASM always
contain exactly one + and no −.
The number of nonzeros in each row and column is odd.
all row and column sums equal 1.
(number of +’s) = n + (number of −’s).
The partial row (column) sums starting from the first or
last entry in a row (column) equal 0 or 1.
Invariant under the dihedral group of order 8 (symmetries
of a square).
⎡ ⎤
+
⎢ ⎢⎢⎢⎢⎢⎣
+ − +
+ − + − +
+ − + − +
⎥
+ − + ⎦
+
Graphs, Designs and Algebraic Combinatorics, 18–21 July 2011

Graphs, Designs and Algebraic Combinatorics, 18–21 July 2011
Enumeration of ASMs
Alternating
Sign Matrices
and Their
Patterns
Richard A.
Brualdi
Alternating
Sign Matrices
(ASMs)
Patterns of
ASMs
Symmetric
ASMs
Additional
Properties of
ASMs
For small n, the number of n × n ASMs is:
1, 2, 7, 42, 429, 7436, . . . .
Major Theorem: The number of n × n ASMs is
1!4!7! · · · (3n − 2)!
n!(n + 1)!(n + 2)! · · · (2n − 1)! .
(Zeilberger 1996 and Kuperberg 1996; conj. of Mills,
Robbins, Rumsey 1983).
1-1 correspondence between ASMs and “square ice”
configurations: a system of water (H 2 O) molecules frozen
in a square lattice.

Graphs, Designs and Algebraic Combinatorics, 18–21 July 2011
Enumeration of ASMs
Alternating
Sign Matrices
and Their
Patterns
Richard A.
Brualdi
Alternating
Sign Matrices
(ASMs)
Patterns of
ASMs
Symmetric
ASMs
Additional
Properties of
ASMs
For small n, the number of n × n ASMs is:
1, 2, 7, 42, 429, 7436, . . . .
Major Theorem: The number of n × n ASMs is
1!4!7! · · · (3n − 2)!
n!(n + 1)!(n + 2)! · · · (2n − 1)! .
(Zeilberger 1996 and Kuperberg 1996; conj. of Mills,
Robbins, Rumsey 1983).
1-1 correspondence between ASMs and “square ice”
configurations: a system of water (H 2 O) molecules frozen
in a square lattice.

Graphs, Designs and Algebraic Combinatorics, 18–21 July 2011
Enumeration of ASMs
Alternating
Sign Matrices
and Their
Patterns
Richard A.
Brualdi
Alternating
Sign Matrices
(ASMs)
Patterns of
ASMs
Symmetric
ASMs
Additional
Properties of
ASMs
For small n, the number of n × n ASMs is:
1, 2, 7, 42, 429, 7436, . . . .
Major Theorem: The number of n × n ASMs is
1!4!7! · · · (3n − 2)!
n!(n + 1)!(n + 2)! · · · (2n − 1)! .
(Zeilberger 1996 and Kuperberg 1996; conj. of Mills,
Robbins, Rumsey 1983).
1-1 correspondence between ASMs and “square ice”
configurations: a system of water (H 2 O) molecules frozen
in a square lattice.

Square Ice I
Alternating
Sign Matrices
and Their
Patterns
Richard A.
Brualdi
Alternating
Sign Matrices
(ASMs)
Patterns of
ASMs
Symmetric
ASMs
Additional
Properties of
ASMs
There are oxygen atoms at each vertex of an n × n lattice, with
hydrogen atoms between successive oxygen atoms in a row or
column, and on either vertical side of the lattice, but not on
the two horizontal sides. E.G. n = 4:
H O H O H O H O H
H H H H
H O H O H O H O H
H H H H
H O H O H O H O H
H H H H
H O H O H O H O H
.
Each O is to be attached to two Hs to get a water molecule
H 2 O, where no two oxygen atoms are attached to any of the
same hydrogen atoms. There are six possible configurations in
which an oxygen atom can be attached to two hydrogen atoms:
Graphs, Designs and Algebraic Combinatorics, 18–21 July 2011

Graphs, Designs and Algebraic Combinatorics, 18–21 July 2011
Square Ice II
Alternating
Sign Matrices
and Their
Patterns
Richard A.
Brualdi
Alternating
Sign Matrices
(ASMs)
Patterns of
ASMs
Symmetric
ASMs
Additional
Properties of
ASMs
H ← O → H
H
↑
H ← O
H
↑
O
↓
H
H
↑
O → H
O → H
↓
H
H ← O
↓
H
The top two configurations correspond to 1 and −1,
respectively. The other four configurations correspond to 0.
.

Square Ice III
Alternating
Sign Matrices
and Their
Patterns
Richard A.
Brualdi
Alternating
Sign Matrices
(ASMs)
Patterns of
ASMs
Symmetric
ASMs
Additional
Properties of
ASMs
(n = 4)
H ← O H ← O H ← O → H O → H
↓ ↓ ↓
H H H H
↑
H ← O → H O → H O H ← O → H
↓
↓
H H H H
↑
↑
H ← O H ← O → H O → H O → H
↓
H H H H
↑ ↑ ↑
H ← O H ← O H ← O → H O → H
and this corresponds to the ASM
2
6
4
Graphs, Designs and Algebraic Combinatorics, 18–21 July 2011
0 0 1 0
1 0 −1 1
0 1 0 0
0 0 1 0
3
7
5 .

Graphs, Designs and Algebraic Combinatorics, 18–21 July 2011
Patterns of n × n ASMs
Alternating
Sign Matrices
and Their
Patterns
Richard A.
Brualdi
Alternating
Sign Matrices
(ASMs)
Patterns of
ASMs
Symmetric
ASMs
Additional
Properties of
ASMs
An n × n ASM A has a decomposition of the form
A = A 1 − A 2
where A 1 and A 2 are (0, 1)-matrices (or (0, +)-matrices), e.g.,
⎡
⎤ ⎡
⎤ ⎡
⎤
0 1 0 0 0 1 0 0 0 0 0 0
⎢ 1 −1 1 0
⎥
⎣ 0 0 0 1 ⎦ = ⎢ 1 0 1 0
⎥
⎣ 0 0 0 1 ⎦ − ⎢ 0 1 0 0
⎥
⎣ 0 0 0 0 ⎦ .
0 1 0 0 0 1 0 0 0 0 0 0
The pattern of an ASM is the (0, 1)-matrix (or (0, +)-matrix)
A 1 + A 2 . In the example, the pattern is
⎡
⎤ ⎡
⎤ ⎡
⎤
0 1 0 0 0 + 0 0
+
⎢ 1 1 1 0
⎥
⎣ 0 0 0 1 ⎦ → ⎢ + + + 0
⎥
⎣ 0 0 0 + ⎦ → ⎢ + + +
⎥
⎣
+ ⎦
0 1 0 0 0 + 0 0
+

Graphs, Designs and Algebraic Combinatorics, 18–21 July 2011
Patterns of n × n ASMs
Alternating
Sign Matrices
and Their
Patterns
Richard A.
Brualdi
Alternating
Sign Matrices
(ASMs)
Patterns of
ASMs
Symmetric
ASMs
Additional
Properties of
ASMs
An n × n ASM A has a decomposition of the form
A = A 1 − A 2
where A 1 and A 2 are (0, 1)-matrices (or (0, +)-matrices), e.g.,
⎡
⎤ ⎡
⎤ ⎡
⎤
0 1 0 0 0 1 0 0 0 0 0 0
⎢ 1 −1 1 0
⎥
⎣ 0 0 0 1 ⎦ = ⎢ 1 0 1 0
⎥
⎣ 0 0 0 1 ⎦ − ⎢ 0 1 0 0
⎥
⎣ 0 0 0 0 ⎦ .
0 1 0 0 0 1 0 0 0 0 0 0
The pattern of an ASM is the (0, 1)-matrix (or (0, +)-matrix)
A 1 + A 2 . In the example, the pattern is
⎡
⎤ ⎡
⎤ ⎡
⎤
0 1 0 0 0 + 0 0
+
⎢ 1 1 1 0
⎥
⎣ 0 0 0 1 ⎦ → ⎢ + + + 0
⎥
⎣ 0 0 0 + ⎦ → ⎢ + + +
⎥
⎣
+ ⎦
0 1 0 0 0 + 0 0
+

Graphs, Designs and Algebraic Combinatorics, 18–21 July 2011
ASMs are ASBGs
Alternating
Sign Matrices
and Their
Patterns
Richard A.
Brualdi
Alternating
Sign Matrices
(ASMs)
Patterns of
ASMs
Symmetric
ASMs
An n × n ASM is an Alternating Sign Bipartite Graph, an
ASBG: A bipartite graph G ⊆ K n,n in which the vertices in
each part of the bipartition are ordered as 1, 2, . . . , n where at
each vertex u the signs of the edges at u alternate
+, −, +, · · · , −, + considering the edges at u in the given
ordering of the vertices in the part not containing u. For
example, with red designating a negative edge,
1
1
Additional
Properties of
ASMs
2
3
4
2
3
4
⎡
⎢
⎣
+
+ − +
+ − +
+
⎤
⎥
⎦

Graphs, Designs and Algebraic Combinatorics, 18–21 July 2011
ASMs are ASBGs
Alternating
Sign Matrices
and Their
Patterns
Richard A.
Brualdi
Alternating
Sign Matrices
(ASMs)
Patterns of
ASMs
Symmetric
ASMs
An n × n ASM is an Alternating Sign Bipartite Graph, an
ASBG: A bipartite graph G ⊆ K n,n in which the vertices in
each part of the bipartition are ordered as 1, 2, . . . , n where at
each vertex u the signs of the edges at u alternate
+, −, +, · · · , −, + considering the edges at u in the given
ordering of the vertices in the part not containing u. For
example, with red designating a negative edge,
1
1
Additional
Properties of
ASMs
2
3
4
2
3
4
⎡
⎢
⎣
+
+ − +
+ − +
+
⎤
⎥
⎦

Graphs, Designs and Algebraic Combinatorics, 18–21 July 2011
ASMs are ASBGs
Alternating
Sign Matrices
and Their
Patterns
Richard A.
Brualdi
Alternating
Sign Matrices
(ASMs)
Patterns of
ASMs
Symmetric
ASMs
An n × n ASM is an Alternating Sign Bipartite Graph, an
ASBG: A bipartite graph G ⊆ K n,n in which the vertices in
each part of the bipartition are ordered as 1, 2, . . . , n where at
each vertex u the signs of the edges at u alternate
+, −, +, · · · , −, + considering the edges at u in the given
ordering of the vertices in the part not containing u. For
example, with red designating a negative edge,
1
1
Additional
Properties of
ASMs
2
3
4
2
3
4
⎡
⎢
⎣
+
+ − +
+ − +
+
⎤
⎥
⎦

Graphs, Designs and Algebraic Combinatorics, 18–21 July 2011
Properties of n × n ASM Patterns
Alternating
Sign Matrices
and Their
Patterns
Richard A.
Brualdi
Alternating
Sign Matrices
(ASMs)
Patterns of
ASMs
Symmetric
ASMs
Additional
Properties of
ASMs
Let R = (r 1 , r 2 , . . . , r n ) and S = (s 1 , s 2 , . . . , s n ) be the row sum
and column sum vectors of the pattern of an n × n ASM.
(1) r 1 + r 2 + · · · + r n = s 1 + s 2 + · · · + s n .
(2) The r i and s j are odd with r 1 = r n = s 1 = s n = 1.
(3) (1, 1, . . . , 1) ≤ R, S ≤ (1, 3, 5, . . . , 5, 3, 1) (entrywise).
(First row has one 1 and no −1’s. So second row has at
most one −1 (below the 1 in row 1) and so at most two
1’s. The third row has at most two −1’s and at most
three 1’s, ... . Similarly, starting from the last row.)
Question: Do (1), (2), and (3) characterize R and S for ASM
patterns?

Graphs, Designs and Algebraic Combinatorics, 18–21 July 2011
Properties of n × n ASM Patterns
Alternating
Sign Matrices
and Their
Patterns
Richard A.
Brualdi
Alternating
Sign Matrices
(ASMs)
Patterns of
ASMs
Symmetric
ASMs
Additional
Properties of
ASMs
Let R = (r 1 , r 2 , . . . , r n ) and S = (s 1 , s 2 , . . . , s n ) be the row sum
and column sum vectors of the pattern of an n × n ASM.
(1) r 1 + r 2 + · · · + r n = s 1 + s 2 + · · · + s n .
(2) The r i and s j are odd with r 1 = r n = s 1 = s n = 1.
(3) (1, 1, . . . , 1) ≤ R, S ≤ (1, 3, 5, . . . , 5, 3, 1) (entrywise).
(First row has one 1 and no −1’s. So second row has at
most one −1 (below the 1 in row 1) and so at most two
1’s. The third row has at most two −1’s and at most
three 1’s, ... . Similarly, starting from the last row.)
Question: Do (1), (2), and (3) characterize R and S for ASM
patterns?

Graphs, Designs and Algebraic Combinatorics, 18–21 July 2011
Properties of n × n ASM Patterns
Alternating
Sign Matrices
and Their
Patterns
Richard A.
Brualdi
Alternating
Sign Matrices
(ASMs)
Patterns of
ASMs
Symmetric
ASMs
Additional
Properties of
ASMs
Let R = (r 1 , r 2 , . . . , r n ) and S = (s 1 , s 2 , . . . , s n ) be the row sum
and column sum vectors of the pattern of an n × n ASM.
(1) r 1 + r 2 + · · · + r n = s 1 + s 2 + · · · + s n .
(2) The r i and s j are odd with r 1 = r n = s 1 = s n = 1.
(3) (1, 1, . . . , 1) ≤ R, S ≤ (1, 3, 5, . . . , 5, 3, 1) (entrywise).
(First row has one 1 and no −1’s. So second row has at
most one −1 (below the 1 in row 1) and so at most two
1’s. The third row has at most two −1’s and at most
three 1’s, ... . Similarly, starting from the last row.)
Question: Do (1), (2), and (3) characterize R and S for ASM
patterns?

Graphs, Designs and Algebraic Combinatorics, 18–21 July 2011
Properties of n × n ASM Patterns
Alternating
Sign Matrices
and Their
Patterns
Richard A.
Brualdi
Alternating
Sign Matrices
(ASMs)
Patterns of
ASMs
Symmetric
ASMs
Additional
Properties of
ASMs
Let R = (r 1 , r 2 , . . . , r n ) and S = (s 1 , s 2 , . . . , s n ) be the row sum
and column sum vectors of the pattern of an n × n ASM.
(1) r 1 + r 2 + · · · + r n = s 1 + s 2 + · · · + s n .
(2) The r i and s j are odd with r 1 = r n = s 1 = s n = 1.
(3) (1, 1, . . . , 1) ≤ R, S ≤ (1, 3, 5, . . . , 5, 3, 1) (entrywise).
(First row has one 1 and no −1’s. So second row has at
most one −1 (below the 1 in row 1) and so at most two
1’s. The third row has at most two −1’s and at most
three 1’s, ... . Similarly, starting from the last row.)
Question: Do (1), (2), and (3) characterize R and S for ASM
patterns?

An Example
Alternating
Sign Matrices
and Their
Patterns
Richard A.
Brualdi
Alternating
Sign Matrices
(ASMs)
Patterns of
ASMs
Symmetric
ASMs
Additional
Properties of
ASMs
(n = 7): Let R = (1, 3, 5, 5, 3, 1, 1) and S = (1, 3, 3, 7, 3, 1, 1).
Then all components are odd,
1 + 3 + 5 + 5 + 3 + 1 + 1 = 19 = 1 + 3 + 3 + 7 + 3 + 1 + 1,
and
(1, 1, 1, 1, 1, 1, 1) ≤ R, S ≤ (1, 3, 5, 7, 5, 3, 1).
But there can be no 7 × 7 ASM whose pattern has row and
column sum vectors R and S, since s 4 = 7 implies that r 6 = 3.
Other necessary conditions result in this way:
r i − 1
2
r i − 1
2
≤ |{j : s j ≥ 3}| for each i, in fact
+ r i+1 − 1
2
Graphs, Designs and Algebraic Combinatorics, 18–21 July 2011
≤ |{j : s j ≥ 3}|.

An Example
Alternating
Sign Matrices
and Their
Patterns
Richard A.
Brualdi
Alternating
Sign Matrices
(ASMs)
Patterns of
ASMs
Symmetric
ASMs
Additional
Properties of
ASMs
(n = 7): Let R = (1, 3, 5, 5, 3, 1, 1) and S = (1, 3, 3, 7, 3, 1, 1).
Then all components are odd,
1 + 3 + 5 + 5 + 3 + 1 + 1 = 19 = 1 + 3 + 3 + 7 + 3 + 1 + 1,
and
(1, 1, 1, 1, 1, 1, 1) ≤ R, S ≤ (1, 3, 5, 7, 5, 3, 1).
But there can be no 7 × 7 ASM whose pattern has row and
column sum vectors R and S, since s 4 = 7 implies that r 6 = 3.
Other necessary conditions result in this way:
r i − 1
2
r i − 1
2
≤ |{j : s j ≥ 3}| for each i, in fact
+ r i+1 − 1
2
Graphs, Designs and Algebraic Combinatorics, 18–21 July 2011
≤ |{j : s j ≥ 3}|.

An Example
Alternating
Sign Matrices
and Their
Patterns
Richard A.
Brualdi
Alternating
Sign Matrices
(ASMs)
Patterns of
ASMs
Symmetric
ASMs
Additional
Properties of
ASMs
(n = 7): Let R = (1, 3, 5, 5, 3, 1, 1) and S = (1, 3, 3, 7, 3, 1, 1).
Then all components are odd,
1 + 3 + 5 + 5 + 3 + 1 + 1 = 19 = 1 + 3 + 3 + 7 + 3 + 1 + 1,
and
(1, 1, 1, 1, 1, 1, 1) ≤ R, S ≤ (1, 3, 5, 7, 5, 3, 1).
But there can be no 7 × 7 ASM whose pattern has row and
column sum vectors R and S, since s 4 = 7 implies that r 6 = 3.
Other necessary conditions result in this way:
r i − 1
2
r i − 1
2
≤ |{j : s j ≥ 3}| for each i, in fact
+ r i+1 − 1
2
Graphs, Designs and Algebraic Combinatorics, 18–21 July 2011
≤ |{j : s j ≥ 3}|.

Graphs, Designs and Algebraic Combinatorics, 18–21 July 2011
Row or Column Sum Vector of ASMs
Alternating
Sign Matrices
and Their
Patterns
Richard A.
Brualdi
Alternating
Sign Matrices
(ASMs)
Patterns of
ASMs
Symmetric
ASMs
Additional
Properties of
ASMs
Theorem: If R = (r 1 , r 2 , . . . , r n ) satisfies
r 1 , r 2 , . . . , r n are odd.
(1, 1, . . . , 1) ≤ (r 1 , r 2 , . . . , r n ) ≤ (1, 3, 5, . . . , 5, 3, 1)
(entrywise), in particular, r 1 = r n = 1,
then R is the row sum vector of the pattern of some n × n
ASM.
Similarly for the column sum vector.

The Diamond ASM: max number of nonzeros
R = S = (1, 3, 5, . . . , 5, 3, 1)
Graphs, Designs and Algebraic Combinatorics, 18–21 July 2011
Alternating
Sign Matrices
and Their
Patterns
Richard A.
Brualdi
Alternating
Sign Matrices
(ASMs)
Patterns of
ASMs
Symmetric
ASMs
Additional
Properties of
ASMs
⎡
⎢
⎣
⎡
⎢
⎣
+
+ − +
+ − + − +
+ − + − +
+ − +
+
+
+ − +
+ − + − +
+ − + − + − +
+ − + − +
+ − +
+
⎤
⎥
⎦
⎤
⎥
⎦
(n even; n2
2 nonzeros)
(n odd : n2 + 1
2
nonzeros).

The Diamond ASM: max number of nonzeros
R = S = (1, 3, 5, . . . , 5, 3, 1)
Graphs, Designs and Algebraic Combinatorics, 18–21 July 2011
Alternating
Sign Matrices
and Their
Patterns
Richard A.
Brualdi
Alternating
Sign Matrices
(ASMs)
Patterns of
ASMs
Symmetric
ASMs
Additional
Properties of
ASMs
⎡
⎢
⎣
⎡
⎢
⎣
+
+ − +
+ − + − +
+ − + − +
+ − +
+
+
+ − +
+ − + − +
+ − + − + − +
+ − + − +
+ − +
+
⎤
⎥
⎦
⎤
⎥
⎦
(n even; n2
2 nonzeros)
(n odd : n2 + 1
2
nonzeros).

Graphs, Designs and Algebraic Combinatorics, 18–21 July 2011
Submatrices of ASMs
Alternating
Sign Matrices
and Their
Patterns
Richard A.
Brualdi
Alternating
Sign Matrices
(ASMs)
Patterns of
ASMs
Symmetric
ASMs
Additional
Properties of
ASMs
No restrictions whatsover: Any k × l (0, +1, −1)-matrix is a
submatrix of some ASM.
⎡
⎤
+ +
For example, starting with the 3 × 3 matrix ⎣ − + ⎦, we
− −
can get the ASM
⎡
⎤
+
+ − +
+
+ − +
.
+ − + − +
+
⎢
⎥
⎣
+
⎦
+

Graphs, Designs and Algebraic Combinatorics, 18–21 July 2011
Submatrices of ASMs
Alternating
Sign Matrices
and Their
Patterns
Richard A.
Brualdi
Alternating
Sign Matrices
(ASMs)
Patterns of
ASMs
Symmetric
ASMs
Additional
Properties of
ASMs
No restrictions whatsover: Any k × l (0, +1, −1)-matrix is a
submatrix of some ASM.
⎡
⎤
+ +
For example, starting with the 3 × 3 matrix ⎣ − + ⎦, we
− −
can get the ASM
⎡
⎤
+
+ − +
+
+ − +
.
+ − + − +
+
⎢
⎥
⎣
+
⎦
+

Graphs, Designs and Algebraic Combinatorics, 18–21 July 2011
Term Rank of ASMs
Alternating
Sign Matrices
and Their
Patterns
Richard A.
Brualdi
Alternating
Sign Matrices
(ASMs)
Patterns of
ASMs
Symmetric
ASMs
Additional
Properties of
ASMs
Term rank ρ(A) of a matrix is the maximum number of
nonzeros, no two of which are from the same row and column.
By König-Egérvary, ρ(A) is the minimum number of rows and
columns containing all the nonzeros of A.
The term rank of an n × n ASM A satisfies
⌈
ρ(A) ≥ 2 √ ⌉
n + 1 − 1 ,
and equality can occur for all n.

Graphs, Designs and Algebraic Combinatorics, 18–21 July 2011
Term Rank of ASMs
Alternating
Sign Matrices
and Their
Patterns
Richard A.
Brualdi
Alternating
Sign Matrices
(ASMs)
Patterns of
ASMs
Symmetric
ASMs
Additional
Properties of
ASMs
Term rank ρ(A) of a matrix is the maximum number of
nonzeros, no two of which are from the same row and column.
By König-Egérvary, ρ(A) is the minimum number of rows and
columns containing all the nonzeros of A.
The term rank of an n × n ASM A satisfies
⌈
ρ(A) ≥ 2 √ ⌉
n + 1 − 1 ,
and equality can occur for all n.

Minimum Term Rank Example
n = 14, ρ = ⌈ 2 · √15
− 2 ⌉ = 6
Alternating
Sign Matrices
and Their
Patterns
Richard A.
Brualdi
Alternating
Sign Matrices
(ASMs)
Patterns of
ASMs
Symmetric
ASMs
Additional
Properties of
ASMs
⎡
⎢
⎣
+
+
+
+ − + − + − +
+
+
+
+ − + − + − +
+
+
+
+ − + − + 0
+
+
Graphs, Designs and Algebraic Combinatorics, 18–21 July 2011
⎤
⎥
⎦

Graphs, Designs and Algebraic Combinatorics, 18–21 July 2011
Alternating Sign Graphs
Alternating
Sign Matrices
and Their
Patterns
Richard A.
Brualdi
Alternating
Sign Matrices
(ASMs)
Patterns of
ASMs
Symmetric
ASMs
Additional
Properties of
ASMs
Recall the diamond ASMs:
⎡
⎡
⎤
+
+ − +
+ − + − +
⎢ + − + − +
,
⎥
⎣ + − + ⎦
⎢
⎣
+
+
+ − +
+ − + − +
+ − + − + − +
+ − + − +
+ − +
+
Symmetric matrices, correspond to alternating sign graphs,
with possible loops: linearly ordered vertices with signed
edges such that: If i is any vertex and {j 1 , j 2 , . . . , j k } are the
vertices joined to i by an edge (k is odd) where
1 ≤ j 1 < j 2 < · · · < j k ≤ n, then
{i, j 1 , } + , {i, j 3 } + , . . . , {i, j k } + &{i, j 2 } − , {i, j 4 } − , . . . .{i, j k−1 } − .
⎤
⎥
⎦

Graphs, Designs and Algebraic Combinatorics, 18–21 July 2011
Zero Diagonal Symmetric ASMs: Loopless ASGs
Alternating
Sign Matrices
and Their
Patterns
Richard A.
Brualdi
Alternating
Sign Matrices
(ASMs)
Patterns of
ASMs
Symmetric
ASMs
Additional
Properties of
ASMs
G(A) a loopless ASG of order n, corresponding to an n × n
symmetric ASM A with zero diagonal: All degrees are odd and
sum to 2 × the number of edges. So n must be even.
What is the maximum number of edges in a loopless ASG
of order n, the maximum number of nonzeros above the
main diagonal in an n × n symmetric ASM with only zeros
on the main diagonal?
The n × n diamond ASMs have the largest number of
nonzeros, and while they are symmetric, there are nonzeros on
the main diagonal.

Graphs, Designs and Algebraic Combinatorics, 18–21 July 2011
Hollowed-diamond ASMs, n ≡ 0 mod 4, n = 4k
Alternating
Sign Matrices
and Their
Patterns
Richard A.
Brualdi
Alternating
Sign Matrices
(ASMs)
Patterns of
ASMs
Symmetric
ASMs
Additional
Properties of
ASMs
n ≡ 0 mod 4, n = 4k. Partition the rows and columns into sets
of size k, 2k, k, as shown for n = 8 and k = 2.
⎡
⎢
⎣
+
+ − +
+ − + − +
+ − + − + − +
+ − + − + − +
+ − + − +
+ − +
+
There are k 2 × 2 diagonal blocks equal to
⎤
⎥
⎦
[ + −
− +
]
.

Graphs, Designs and Algebraic Combinatorics, 18–21 July 2011
Hollowed-diamond ASMs, n ≡ 0 mod 4, n = 4k
Alternating
Sign Matrices
and Their
Patterns
Richard A.
Brualdi
Alternating
Sign Matrices
(ASMs)
Patterns of
ASMs
Symmetric
ASMs
Additional
Properties of
ASMs
Replacing those 2 × 2 diagonal blocks by zero matrices we get
a symmetric ASM with all zeros on the main diagonal:
⎡
⎢
⎣
+
+ − +
0 0 + − +
+ 0 0 − + − +
+ − + − 0 0 +
+ − + 0 0
+ − +
+
⎤
⎥
⎦

Graphs, Designs and Algebraic Combinatorics, 18–21 July 2011
Hollowed-diamond ASMs, n ≡ 0 mod 4, n = 4k
Alternating
Sign Matrices
and Their
Patterns
Richard A.
Brualdi
Alternating
Sign Matrices
(ASMs)
Patterns of
ASMs
Symmetric
ASMs
Additional
Properties of
ASMs
Reversing the rows of a diamond ASM, we get an equivalent
ASM:
⎡
D 8 =
⎢
⎣
+
+ − +
+ − + − +
+ − + − + − +
+ − + − + − +
+ − + − +
+ − +
+
and “hollowing it out”, we get another n × n symmetric ASM
with all zeros on the main diagonal.
⎤
⎥
⎦

Graphs, Designs and Algebraic Combinatorics, 18–21 July 2011
Hollowed-diamond ASMs, n ≡ 0 mod 4, n = 4k
Alternating
Sign Matrices
and Their
Patterns
Richard A.
Brualdi
Alternating
Sign Matrices
(ASMs)
Patterns of
ASMs
Symmetric
ASMs
Additional
Properties of
ASMs
⎡
⎢
⎣
+
+ − +
+ 0 0 − +
+ − 0 0 + − +
+ − + 0 0 − +
+ − 0 0 +
+ − +
+
⎤
= D 8.
∗ ⎥
⎦
The two ASMs obtained in this way are not equivalent under
the action of the dihedral group acting on a square.

Graphs, Designs and Algebraic Combinatorics, 18–21 July 2011
Hollowed-diamond ASMs, n ≡ 0 mod 4, n = 4k
Alternating
Sign Matrices
and Their
Patterns
Richard A.
Brualdi
Alternating
Sign Matrices
(ASMs)
Patterns of
ASMs
Symmetric
ASMs
Additional
Properties of
ASMs
Another n × n symmetric ASM with zero diagonal with the
same number of nonzero entries, namely n2 −2n
4
= 4k 2 − 2k:
Start with a near-diamond ASM E n of order n = 4k with four
fewer nonzero entries than D n and replacing k − 1 disjoint
2 × 2 principal submatrices
[ ] − +
+ −
with zero matrices to get a symmetric ASM E ∗ n with zero
diagonal, a hollowed-near-diamond ASM.

Graphs, Designs and Algebraic Combinatorics, 18–21 July 2011
Hollowed-diamond ASMs, n ≡ 0 mod 4, n = 4k
Alternating
Sign Matrices
and Their
Patterns
Richard A.
Brualdi
Alternating
Sign Matrices
(ASMs)
Patterns of
ASMs
Symmetric
ASMs
Additional
Properties of
ASMs
We illustrate this construction, again for n = 8.
⎡
E 8 =
⎢
⎣
+
+ − +
+ − + − +
+ − + − +
+ − + − +
+ − + − +
+ − +
+
⎤
⎥
⎦

Graphs, Designs and Algebraic Combinatorics, 18–21 July 2011
Hollowed-diamond ASMs, n ≡ 0 mod 4, n = 4k
Alternating
Sign Matrices
and Their
Patterns
Richard A.
Brualdi
Alternating
Sign Matrices
(ASMs)
Patterns of
ASMs
Symmetric
ASMs
Additional
Properties of
ASMs
⎡
⎢
⎣
+
+ − +
+ − + − +
+ 0 0 − +
+ − 0 0 +
+ − + − +
+ − +
+
⎤
= E8 ∗ .
⎥
⎦

Hollowed-diamond ASMs, n ≡ 2 mod 4
Alternating
Sign Matrices
and Their
Patterns
Richard A.
Brualdi
Alternating
Sign Matrices
(ASMs)
Patterns of
ASMs
Symmetric
ASMs
Additional
Properties of
ASMs
n ≡ 2 mod 4, n = 4k + 2. Partition the rows and columns into
sets of size k + 1, 2k, k + 1, as shown for n = 10 and k = 2.
⎡
⎤
+
+ − +
+ − + − +
+ − + − + − +
D 10 =
+ − + − + − + − +
+ − + − + − + − +
.
+ − + − + − +
+ − + − +
⎢
⎥
⎣
+ − +
⎦
+
There are k 2 × 2 diagonal blocks equal to
Graphs, Designs and Algebraic Combinatorics, 18–21 July 2011
[ − +
+ −
]
.

Graphs, Designs and Algebraic Combinatorics, 18–21 July 2011
Hollowed-diamond ASMs, n ≡ 2 mod 4
Alternating
Sign Matrices
and Their
Patterns
Richard A.
Brualdi
Alternating
Sign Matrices
(ASMs)
Patterns of
ASMs
Symmetric
ASMs
Additional
Properties of
ASMs
Replacing those 2 × 2 diagonal blocks by zero matrices we get
a symmetric ASM with all zeros on the main diagonal:
⎡
⎤
+
+ − +
+ − + − +
+ 0 0 − + − +
D10 ∗ =
+ − 0 0 + − + − +
+ − + − + 0 0 − +
.
+ − + − 0 0 +
+ − + − +
⎢
⎥
⎣
+ − +
⎦
+

Graphs, Designs and Algebraic Combinatorics, 18–21 July 2011
Maximal ASMs
Alternating
Sign Matrices
and Their
Patterns
An extension of an n × n ASM A = [a ij ] is an n × n ASM
B = [b ij ] such that B ≠ A and
Richard A.
Brualdi
a ij ≠ 0 implies b ij = a ij
(1 ≤ i, j ≤ n).
Alternating
Sign Matrices
(ASMs)
Patterns of
ASMs
Symmetric
ASMs
Additional
Properties of
ASMs
Examples:
⎡
⎢
⎣
⎡
⎢
⎣
0 1 0 0 0
0 0 0 1 0
1 0 0 0 0
0 0 0 0 1
0 0 1 0 0
1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1
⎤
⎥
⎦
⎤ ⎡
⎥
⎦ is not: ⎢
⎣
is maximal .
0 1 0 0 0
0 0 0 1 0
1 −1 1 0 0
0 1 −1 0 1
0 0 1 0 0
⎤
⎥
⎦ .

Graphs, Designs and Algebraic Combinatorics, 18–21 July 2011
Maximal ASMs
Alternating
Sign Matrices
and Their
Patterns
Richard A.
Brualdi
Alternating
Sign Matrices
(ASMs)
Patterns of
ASMs
Symmetric
ASMs
Additional
Properties of
ASMs
An elementary ASM extension of an ASM is an ASM
extension obtained by replacing a 2 × 2 zero submatrix by
[ ]
1 −1
T = ±
.
−1 1
Recall that permutation matrices are ASMs. Then:
A permutation matrix is not a maximal ASM iff it has an
elementary ASM extension.

Graphs, Designs and Algebraic Combinatorics, 18–21 July 2011
Maximal ASMs
Alternating
Sign Matrices
and Their
Patterns
Richard A.
Brualdi
Alternating
Sign Matrices
(ASMs)
Patterns of
ASMs
Symmetric
ASMs
Additional
Properties of
ASMs
[ ]
1 −1
It is sometimes possible to add T = ±
to a 2 × 2
−1 1
nonzero submatrix of an ASM with the result being an ASM.
E.G.
⎡
⎤
⎤
⎢
⎣
0 0 +1 0 0
0 +1 0 0 0
+1 0 −1 0 +1
0 0 0 +1 0
0 0 +1 0 0
⎡
⎢
⎣
⎡
⎥
⎦ + ⎢
⎣
0 0 +1 0 0
0 +1 −1 +1 0
+1 0 0 −1 +1
0 0 0 +1 0
0 0 +1 0 0
0 0 0 0 0
0 0 −1 +1 0
0 0 +1 −1 0
0 0 0 0 0
0 0 0 0 0
⎤
⎥
⎦
⎥
⎦ =

Graphs, Designs and Algebraic Combinatorics, 18–21 July 2011
Generating ASMs
Alternating
Sign Matrices
and Their
Patterns
Richard A.
Brualdi
Alternating
Sign Matrices
(ASMs)
Patterns of
ASMs
Symmetric
ASMs
Additional
Properties of
ASMs
[ ]
1 −1
Call this operation of adding T = ±
to a 2 × 2
−1 1
(not necessarily) nonzero submatrix of an ASM an ASM
interchange. ASM interchanges generalize transpositions of
permutations. Basically, the reason is that
[ 1 0
0 1
]
−
[
1 −1
−1 1
]
=
[ 0 1
1 0
Any n × n ASM can be gotten from I n by a sequence of
ASM interchanges.
]
.

Graphs, Designs and Algebraic Combinatorics, 18–21 July 2011
Generating ASMs
Alternating
Sign Matrices
and Their
Patterns
Richard A.
Brualdi
Alternating
Sign Matrices
(ASMs)
Patterns of
ASMs
Symmetric
ASMs
Additional
Properties of
ASMs
[ ]
1 −1
Call this operation of adding T = ±
to a 2 × 2
−1 1
(not necessarily) nonzero submatrix of an ASM an ASM
interchange. ASM interchanges generalize transpositions of
permutations. Basically, the reason is that
[ 1 0
0 1
]
−
[
1 −1
−1 1
]
=
[ 0 1
1 0
Any n × n ASM can be gotten from I n by a sequence of
ASM interchanges.
]
.

Graphs, Designs and Algebraic Combinatorics, 18–21 July 2011
Problems
Alternating
Sign Matrices
and Their
Patterns
Richard A.
Brualdi
Alternating
Sign Matrices
(ASMs)
Patterns of
ASMs
Symmetric
ASMs
Additional
Properties of
ASMs
Characterize maximal ASMs.
We know how to characterize maximal ASMs that are
permutation matrices, but not all ASMs.
Given integers k and l, we know that every k × l
(0, 1, −1)-matrix is a submatrix of an n × n ASM for
some n. What is the minimum n over all such k × l
matrices?
Reference: Patterns of alternating sign matrices by
R.A. Brualdi, K,P. Kiernan, S.A. Meyer, and M.W. Schroeder,
submitted.