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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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**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**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 L1 = W1 > L2 = W2 > L3 = W3 > … width length**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**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**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**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**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**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.**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**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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