1 0 –1 1 0 –1 0 1 1979, Andrews counts cyclically symmetric plane partitions.
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Transcript of 1 0 –1 1 0 –1 0 1 1979, Andrews counts cyclically symmetric plane partitions.
![Page 1: 1 0 –1 1 0 –1 0 1 1979, Andrews counts cyclically symmetric plane partitions.](https://reader036.fdocuments.in/reader036/viewer/2022082510/5a4d1b167f8b9ab059991824/html5/thumbnails/1.jpg)
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1979, Andrews counts cyclically symmetric plane partitions
![Page 2: 1 0 –1 1 0 –1 0 1 1979, Andrews counts cyclically symmetric plane partitions.](https://reader036.fdocuments.in/reader036/viewer/2022082510/5a4d1b167f8b9ab059991824/html5/thumbnails/2.jpg)
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1979, Andrews counts cyclically symmetric plane partitions
![Page 3: 1 0 –1 1 0 –1 0 1 1979, Andrews counts cyclically symmetric plane partitions.](https://reader036.fdocuments.in/reader036/viewer/2022082510/5a4d1b167f8b9ab059991824/html5/thumbnails/3.jpg)
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1979, Andrews counts cyclically symmetric plane partitions
![Page 4: 1 0 –1 1 0 –1 0 1 1979, Andrews counts cyclically symmetric plane partitions.](https://reader036.fdocuments.in/reader036/viewer/2022082510/5a4d1b167f8b9ab059991824/html5/thumbnails/4.jpg)
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1979, Andrews counts cyclically symmetric plane partitions
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1979, Andrews counts cyclically symmetric plane partitions
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L1 = W1 > L2 = W2 > L3 = W3 > …
![Page 6: 1 0 –1 1 0 –1 0 1 1979, Andrews counts cyclically symmetric plane partitions.](https://reader036.fdocuments.in/reader036/viewer/2022082510/5a4d1b167f8b9ab059991824/html5/thumbnails/6.jpg)
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1979, Andrews counts descending plane partitions
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L1 > W1 ≥ L2 > W2 ≥ L3 > W3 ≥ …
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6 X 6 ASM DPP with largest part ≤ 6
What are the corresponding 6 subsets of DPP’s?
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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.
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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.
![Page 10: 1 0 –1 1 0 –1 0 1 1979, Andrews counts cyclically symmetric plane partitions.](https://reader036.fdocuments.in/reader036/viewer/2022082510/5a4d1b167f8b9ab059991824/html5/thumbnails/10.jpg)
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Mills, Robbins, & Rumsey proved that # of DPP’s ≤ n with j parts of size n was given by their conjectured formula for ASM’s.
![Page 11: 1 0 –1 1 0 –1 0 1 1979, Andrews counts cyclically symmetric plane partitions.](https://reader036.fdocuments.in/reader036/viewer/2022082510/5a4d1b167f8b9ab059991824/html5/thumbnails/11.jpg)
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Mills, Robbins, & Rumsey proved that # of DPP’s ≤ n with j parts of size n was given by their conjectured formula for ASM’s. Discovered an easier proof of Andrews’ formula, using induction on j and n.
![Page 12: 1 0 –1 1 0 –1 0 1 1979, Andrews counts cyclically symmetric plane partitions.](https://reader036.fdocuments.in/reader036/viewer/2022082510/5a4d1b167f8b9ab059991824/html5/thumbnails/12.jpg)
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Mills, Robbins, & Rumsey proved that # of DPP’s ≤ n with j parts of size n was given by their conjectured formula for ASM’s. 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
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Mills, Robbins, & Rumsey proved that # of DPP’s ≤ n with j parts of size n was given by their conjectured formula for ASM’s. 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!