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1 0 0 –1 1 0 –1 0 1

1 0 0 –1 1 0 –1 0 1

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1 0 0 –1 1 0 –1 0 1

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  1. Totally Symmetric Self-Complementary Plane Partitions 1 0 0 –1 1 0 –1 0 1 1983

  2. Totally Symmetric Self-Complementary Plane Partitions 1 0 0 –1 1 0 –1 0 1

  3. 1 0 0 –1 1 0 –1 0 1

  4. Robbins’ Conjecture: The number of TSSCPP’s in a 2n X 2n X 2n box is 1 0 0 –1 1 0 –1 0 1

  5. Robbins’ Conjecture: The number of TSSCPP’s in a 2n X 2n X 2n box is 1 0 0 –1 1 0 –1 0 1 1989: William Doran shows equivalent to counting lattice paths 1990: John Stembridge represents the counting function as a Pfaffian (built on insights of Gordon and Okada) 1992: George Andrews evaluates the Pfaffian, proves Robbins’ Conjecture

  6. December, 1992 Doron Zeilberger (now at Rutgers University) announces a proof that # of ASM’s of size n equals of TSSCPP’s in box of size 2n. 1 0 0 –1 1 0 –1 0 1

  7. December, 1992 Doron Zeilberger (now at Rutgers University) announces a proof that # of ASM’s of size n equals of TSSCPP’s in box of size 2n. 1 0 0 –1 1 0 –1 0 1 1995 all gaps removed, published as “Proof of the Alternating Sign Matrix Conjecture,” Elect. J. of Combinatorics, 1996.

  8. Zeilberger’s proof is an 84-page tour de force, but it still left open the original conjecture: 1 0 0 –1 1 0 –1 0 1

  9. 1996 Greg Kuperberg (U.C. Davis) announces a simple proof 1 0 0 –1 1 0 –1 0 1 “Another proof of the alternating sign matrix conjecture,” International Mathematics Research Notices

  10. 1996 Greg Kuperberg (U.C. Davis) announces a simple proof 1 0 0 –1 1 0 –1 0 1 “Another proof of the alternating sign matrix conjecture,” International Mathematics Research Notices Physicists have been studying ASM’s for decades, only they call them square ice (aka the six-vertex model ).

  11. H O H O H O H O H O H H H H H H H O H O H O H O H O H H H H H H H O H O H O H O H O H H H H H H H O H O H O H O H O H H H H H H H O H O H O H O H O H 1 0 0 –1 1 0 –1 0 1

  12. 1960’s Rodney Baxter’s Triangle-to-triangle relation 1 0 0 –1 1 0 –1 0 1

  13. 1960’s Rodney Baxter’s Triangle-to-triangle relation 1 0 0 –1 1 0 –1 0 1 1980’s Vladimir Korepin Anatoli Izergin

  14. 1 0 0 –1 1 0 –1 0 1

  15. 1996 Doron Zeilberger uses this determinant to prove the original conjecture 1 0 0 –1 1 0 –1 0 1 “Proof of the refined alternating sign matrix conjecture,” New York Journal of Mathematics

  16. The End 1 0 0 –1 1 0 –1 0 1 (which is really just the beginning)

  17. Margaret Readdy Univ. of Kentucky The End 1 0 0 –1 1 0 –1 0 1 (which is really just the beginning)

  18. Margaret Readdy Univ. of Kentucky The End 1 0 0 –1 1 0 –1 0 1 Anne Schilling Univ. of Calif. Davis

  19. Margaret Readdy Univ. of Kentucky 1 0 0 –1 1 0 –1 0 1 Anne Schilling Univ. of Calif. Davis Mireille Bousquet-Mélou LaBRI, Univ. Bordeaux