Proofs & Confirmations The story of the alternating sign matrix conjecture David M. Bressoud Macalester College Hollins University April 29, 2004

Institute for Defense Analysis Bill Mills Institute for Defense Analysis 1 –1 Howard Rumsey Dave Robbins

Charles L. Dodgson aka Lewis Carroll 1 –1 “Condensation of Determinants,” Proceedings of the Royal Society, London 1866

1 –1

1 –1 A5 = 429

1 –1

1 –1 Monotone Triangle

1 –1 Monotone Triangle

1 2 3 4 5 1 –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

1 2 3 4 5 1 –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

1 2 3 4 5 1 –1 12 13 14 15 23 24 25 34 35 45 123 124 125 134 135 145 234 235 345 2 3 2 4 3 2 5 4 2 1234 1235 1245 1345 2345 12345

1 2 3 4 5 1 –1 12 13 14 15 23 24 25 34 35 45 14 123 124 125 134 135 145 234 235 345 2 3 2 4 3 2 5 4 2 1234 1235 1245 1345 2345 12345

1 2 3 4 5 1 –1 12 13 14 15 23 24 25 34 35 45 7 14 14 7 23 26 14 23 14 7 123 124 125 134 135 145 234 235 345 2 3 2 4 3 2 5 4 2 1234 1235 1245 1345 2345 12345

1 2 3 4 5 1 –1 105 12 13 14 15 23 24 25 34 35 45 7 14 14 7 23 26 14 23 14 7 123 124 125 134 135 145 234 235 345 2 3 2 4 3 2 5 4 2 1234 1235 1245 1345 2345 12345

1 2 3 4 5 1 –1 42 105 135 105 42 12 13 14 15 23 24 25 34 35 45 7 14 14 7 23 26 14 23 14 7 123 124 125 134 135 145 234 235 345 2 3 2 4 3 2 5 4 2 1234 1235 1245 1345 2345 12345

429 1 2 3 4 5 1 –1 42 105 135 105 42 12 13 14 15 23 24 25 34 35 45 7 14 14 7 23 26 14 23 14 7 123 124 125 134 135 145 234 235 345 2 3 2 4 3 2 5 4 2 1234 1235 1245 1345 2345 12345

1 –1 A5 = 429 A10 = 129, 534, 272, 700

1 –1 A5 = 429 A10 = 129, 534, 272, 700 A20 = 1436038934715538200913155682637051204376827212 = 1.43… 1045