Proofs

1
Proofs Confirmations The story of the
alternating sign matrix conjecture
David M. Bressoud Macalester College
Hollins University April 29, 2004
2
Bill Mills
Institute for Defense Analysis
1 0 0 1 1 0 1 0 1
Howard Rumsey
Dave Robbins
3
Charles L. Dodgson aka Lewis Carroll
1 0 0 1 1 0 1 0 1
Condensation of Determinants, Proceedings of
the Royal Society, London 1866
4
1 0 0 1 1 0 1 0 1
5
1 0 0 1 1 0 1 0 1
A5 429
6
1 0 0 1 1 0 1 0 1
7
1 0 0 1 1 0 1 0 1
Monotone Triangle
8
1 0 0 1 1 0 1 0 1
Monotone Triangle
9
1 2 3 4 5
1 0 0 1 1 0 1 0 1
12 13 14 15 23 24 25 34 35
45
123 124 125 134 135 145 234
235 345
1234 1235 1245 1345 2345
12345
10
1 2 3 4 5
1 0 0 1 1 0 1 0 1
12 13 14 15 23 24 25 34 35
45
123 124 125 134 135 145 234
235 345
3
1234 1235 1245 1345 2345
12345
11
1 2 3 4 5
1 0 0 1 1 0 1 0 1
12 13 14 15 23 24 25 34 35
45
123 124 125 134 135 145 234
235 345
3
2
2
4
3
2
5
4
2
1234 1235 1245 1345 2345
12345
12
1 2 3 4 5
1 0 0 1 1 0 1 0 1
12 13 14 15 23 24 25 34 35
45
14
123 124 125 134 135 145 234
235 345
3
2
2
4
3
2
5
4
2
1234 1235 1245 1345 2345
12345
13
1 2 3 4 5
1 0 0 1 1 0 1 0 1
12 13 14 15 23 24 25 34 35
45
14
7
23
14
26
23
7
14
14
7
123 124 125 134 135 145 234
235 345
3
2
2
4
3
2
5
4
2
1234 1235 1245 1345 2345
12345
14
1 2 3 4 5
1 0 0 1 1 0 1 0 1
105
12 13 14 15 23 24 25 34 35
45
14
7
23
14
26
23
7
14
14
7
123 124 125 134 135 145 234
235 345
3
2
2
4
3
2
5
4
2
1234 1235 1245 1345 2345
12345
15
1 2 3 4 5
1 0 0 1 1 0 1 0 1
42
105
135
105
42
12 13 14 15 23 24 25 34 35
45
14
7
23
14
26
23
7
14
14
7
123 124 125 134 135 145 234
235 345
3
2
2
4
3
2
5
4
2
1234 1235 1245 1345 2345
12345
16
429
1 2 3 4 5
1 0 0 1 1 0 1 0 1
42
105
135
105
42
12 13 14 15 23 24 25 34 35
45
14
7
23
14
26
23
7
14
14
7
123 124 125 134 135 145 234
235 345
3
2
2
4
3
2
5
4
2
1234 1235 1245 1345 2345
12345
17
1 0 0 1 1 0 1 0 1
A5 429 A10 129, 534, 272, 700
18
1 0 0 1 1 0 1 0 1
A5 429 A10 129, 534, 272, 700 A20
1436038934715538200913155682637051204376827212
1.43 ?1045