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1 0 0 –1 1 0 –1 0 1

1979, Andrews counts cyclically symmetric plane partitions. 1 0 0 –1 1 0 –1 0 1. 1979, Andrews counts cyclically symmetric plane partitions. 1 0 0 –1 1 0 –1 0 1. 1979, Andrews counts cyclically symmetric plane partitions. 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. 1979, Andrews counts cyclically symmetric plane partitions 1 0 0 –1 1 0 –1 0 1

  2. 1979, Andrews counts cyclically symmetric plane partitions 1 0 0 –1 1 0 –1 0 1

  3. 1979, Andrews counts cyclically symmetric plane partitions 1 0 0 –1 1 0 –1 0 1

  4. 1979, Andrews counts cyclically symmetric plane partitions 1 0 0 –1 1 0 –1 0 1

  5. 1979, Andrews counts cyclically symmetric plane partitions 1 0 0 –1 1 0 –1 0 1 L1 = W1 > L2 = W2 > L3 = W3 > … width length

  6. 1979, Andrews counts descending plane partitions 1 0 0 –1 1 0 –1 0 1 L1 > W1 ≥ L2 > W2 ≥ L3 > W3 ≥ … 6 6 6 4 3 3 3 2 width length

  7. 6 X 6 ASM  DPP with largest part ≤ 6 What are the corresponding 6 subsets of DPP’s? 1 0 0 –1 1 0 –1 0 1 6 6 6 4 3 3 3 2 width length

  8. ASM with 1 at top of first column DPP with no parts of size n. ASM with 1 at top of last column DPP with n–1 parts of size n. 1 0 0 –1 1 0 –1 0 1 6 6 6 4 3 3 3 2 width length

  9. Mills, Robbins, Rumsey Conjecture: # of n by n ASM’s with 1 at top of column j equals # of DPP’s ≤ n with exactly j–1 parts of size n. 1 0 0 –1 1 0 –1 0 1 6 6 6 4 3 3 3 2 width length

  10. Mills, Robbins, & Rumsey proved that # of DPP’s ≤ n with j parts of size n was given by their conjectured formula for ASM’s. 1 0 0 –1 1 0 –1 0 1

  11. Mills, Robbins, & Rumsey proved that # of DPP’s ≤ n with j parts of size n was given by their conjectured formula for ASM’s. 1 0 0 –1 1 0 –1 0 1 Discovered an easier proof of Andrews’ formula, using induction on j and n.

  12. Mills, Robbins, & Rumsey proved that # of DPP’s ≤ n with j parts of size n was given by their conjectured formula for ASM’s. 1 0 0 –1 1 0 –1 0 1 Discovered an easier proof of Andrews’ formula, using induction on j and n. Used this inductive argument to prove Macdonald’s conjecture “Proof of the Macdonald Conjecture,” Inv. Math., 1982

  13. Mills, Robbins, & Rumsey proved that # of DPP’s ≤ n with j parts of size n was given by their conjectured formula for ASM’s. 1 0 0 –1 1 0 –1 0 1 Discovered an easier proof of Andrews’ formula, using induction on j and n. Used this inductive argument to prove Macdonald’s conjecture “Proof of the Macdonald Conjecture,” Inv. Math., 1982 But they still didn’t have a proof of their conjecture!

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