A Baker's dozen of conjectures concerning plane partitions

We prove the Razumov–Stroganov conjecture relating ground state of the $O(1)$ loop model and counting of Fully Packed Loops in the case of certain types of link patterns. The main focus is on link patterns with three series of nested arches, for which we use as key ingredient of the proof a generalization of the MacMahon formula for the number of plane partitions which includes three series of parameters.

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Plane partitions. II. The equivalence of the Bender-Knuth and MacMahon conjectures

Pacific Journal of Mathematics ◽

10.2140/pjm.1977.72.283 ◽

1977 ◽

Vol 72 (2) ◽

pp. 283-291 ◽

Cited By ~ 18

Author(s):

George Andrews

Keyword(s):

Plane Partitions

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Correlation functions for a strongly coupled boson system and plane partitions

Philosophical Transactions of The Royal Society A Mathematical Physical and Engineering Sciences ◽

10.1098/rsta.2010.0322 ◽

2011 ◽

Vol 369 (1939) ◽

pp. 1319-1333 ◽

Cited By ~ 10

Author(s):

Nikolay Bogoliubov ◽

Jussi Timonen

Keyword(s):

Correlation Functions ◽

Finite Size ◽

Temperature Limit ◽

Phase Model ◽

Strong Interactions ◽

Scaling Exponent ◽

Strongly Correlated ◽

Strongly Coupled ◽

Plane Partitions ◽

Hopping Model

A quantum phase model is introduced as a limit for very strong interactions of a strongly correlated q -boson hopping model. The exact solution of the phase model is reviewed, and solutions are also provided for two correlation functions of the model. Explicit expressions, including both amplitude and scaling exponent, are derived for these correlation functions in the low temperature limit. The amplitudes were found to be related to the number of plane partitions contained in boxes of finite size.

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