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Investigation of Cyclically Symmetric Plane Partitions and Ascending Subsets

This study delves into the relationships between cyclically symmetric plane partitions and ascending subsets, exploring various conjectures and proofs regarding their structures. The intricate mathematical patterns and formulas are analyzed in-depth for a comprehensive understanding.

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Investigation of Cyclically Symmetric Plane Partitions and Ascending Subsets

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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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