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1 0 0 –1 1 0 –1 0 1

Percy A. MacMahon. Plane Partition. 1 0 0 –1 1 0 –1 0 1. Work begun in 1897. Plane partition of 75. 6 5 5 4 3 3. 1 0 0 –1 1 0 –1 0 1. # of pp’s of 75 = pp (75). Plane partition of 75. 6 5 5 4 3 3. 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. Percy A. MacMahon Plane Partition 1 0 0 –1 1 0 –1 0 1 Work begun in 1897

  2. Plane partition of 75 6 5 5 4 3 3 1 0 0 –1 1 0 –1 0 1 # of pp’s of 75 = pp(75)

  3. Plane partition of 75 6 5 5 4 3 3 1 0 0 –1 1 0 –1 0 1 # of pp’s of 75 = pp(75) = 37,745,732,428,153

  4. 1 0 0 –1 1 0 –1 0 1

  5. Generating function: 1 0 0 –1 1 0 –1 0 1

  6. 1 0 0 –1 1 0 –1 0 1

  7. 1 0 0 –1 1 0 –1 0 1

  8. 1 0 0 –1 1 0 –1 0 1

  9. 1 0 0 –1 1 0 –1 0 1 0 1 2 3 4 5 j 2(j) 1 5 10 21 26 pp(j) 1 1 3 6 1·10 + 1·5 + 3·1 = 3·pp (3)

  10. 1 0 0 –1 1 0 –1 0 1 0 1 2 3 4 5 j 2(j) 1 5 10 21 26 pp(j) 1 1 3 6 13 1·21 + 1·10 + 3·5 + 6·1 = 4·pp (4)

  11. 1 0 0 –1 1 0 –1 0 1 0 1 2 3 4 5 j 2(j) 1 5 10 21 26 pp(j) 1 1 3 6 13 30 1·26 + 1·21 + 3·10 + 6·5 + 13·1 = 5·pp (5)

  12. 1912 MacMahon proves that the generating function for plane partitions in an n X n X n box is 1 0 0 –1 1 0 –1 0 1 At the same time, he conjectures that the generating function for symmetric plane partitions is

  13. Symmetric Plane Partition 4 3 2 1 1 3 2 2 1 2 2 1 1 1 1 1 0 0 –1 1 0 –1 0 1 “The reader must be warned that, although there is little doubt that this result is correct, … the result has not been rigorously established. … Further investigations in regard to these matters would be sure to lead to valuable work.’’ (1916)

  14. 1971 Basil Gordon (UCLA) proves case for n = infinity 1 0 0 –1 1 0 –1 0 1

  15. 1971 Basil Gordon (UCLA) proves case for n = infinity 1 0 0 –1 1 0 –1 0 1 1977 George Andrews and Ian Macdonald (Univ. London) independently prove general case

  16. Cyclically Symmetric Plane Partition 1 0 0 –1 1 0 –1 0 1

  17. Cyclically Symmetric Plane Partition 1 0 0 –1 1 0 –1 0 1

  18. Cyclically Symmetric Plane Partition 1 0 0 –1 1 0 –1 0 1

  19. Cyclically Symmetric Plane Partition 1 0 0 –1 1 0 –1 0 1

  20. Macdonald’s Conjecture (1979): The generating function for cyclically symmetric plane partitions in B(n,n,n) is 1 0 0 –1 1 0 –1 0 1 “If I had to single out the most interesting open problem in all of enumerative combinatorics, this would be it.” Richard Stanley, review of Symmetric Functions and Hall Polynomials, Bulletin of the AMS, March, 1981.

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