Geodesic
Refined Counting of Fully Packed Loop Configurations
Séminaire lotharingien de combinatoire, Tome 56 (2006-2007)

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We give a generalisation of a conjecture by Propp on a summation formula for fully packed loop configurations. The original conjecture states that the number of configurations in which each external edge is connected to its neighbour is equal to the total number of configurations of size one less. This conjecture was later generalised by Zuber to include more types of configurations. Our conjecture further refines the counting and provides a general framework for some other summation formulas observed by Zuber. It also implies similar summation formulas for half-turn symmetric configurations. 

@article{SLC_2006-2007_56_a4,
     author = {Johan Thapper},
     title = {Refined {Counting} of {Fully} {Packed} {Loop} {Configurations}},
     journal = {S\'eminaire lotharingien de combinatoire},
     publisher = {mathdoc},
     volume = {56},
     year = {2006-2007},
     url = {http://geodesic.mathdoc.fr/item/SLC_2006-2007_56_a4/}
}
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Johan Thapper. Refined Counting of Fully Packed Loop Configurations. Séminaire lotharingien de combinatoire, Tome 56 (2006-2007). http://geodesic.mathdoc.fr/item/SLC_2006-2007_56_a4/