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Proofs & Confirmations The story of the alternating sign matrix conjecture

Proofs & Confirmations The story of the alternating sign matrix conjecture. David M. Bressoud Macalester College. Hollins University April 29, 2004. Bill Mills. Institute for Defense Analysis. 1 0 0 –1 1 0 –1 0 1. Howard Rumsey. Dave Robbins. Charles L. Dodgson

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Proofs & Confirmations The story of the alternating sign matrix conjecture

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  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 2 3 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 2 3 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 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

  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 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

  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 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

  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 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

  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

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