Mathematics – Combinatorics
Scientific paper
2011-11-25
Mathematics
Combinatorics
25 pages, many figures. This article contains some of the work presented at the Fpsac 2010 conference in San Francisco
Scientific paper
Fully Packed Loop configurations (FPLs) are certain configurations on the square grid, naturally refined according to certain link patterns. If $A_X$ is the number of FPLs with link pattern $X$, the Razumov--Stroganov correspondence provides relations between numbers $A_X$ relative to a given grid size. In another line of research, if $X\cup p$ denotes $X$ with $p$ additional nested arches, then $A_{X\cup p}$ was shown to be polynomial in $p$: the proof gives rise to certain configurations of FPLs in a triangle (TFPLs). In this work we investigate these TFPL configurations and their relation to FPLs. We prove certain properties of TFPLs, and enumerate them under special boundary conditions. From this study we deduce a class of linear relations, conjectured by Thapper, between quantities $A_X$ relative to different grid sizes, relations which thus differ from the Razumov--Stroganov ones.
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