Sách A conjectured formula for Fully Packed Loop configurations in a triangle

In the literature generated by the seminal papers and revolving around the so-called Razumov Stroganov (RS) conjecture, it has often been remarked that there are more conjectures than theorems. The present work, sadly, will not help correct this imbalance: it is entirely based on one more conjecture. The latter, however, is of some interest since it connects the two sides of the Razumov–Stroganov conjecture; that is, it expresses the number of Fully Packed Loop configurations (FPLs) in a triangle with certain boundary conditions as the constant term of a quasi-generating function which is closely related to expressions appearing in the context of the Temperley–Lieb(1) (sometimes called O(1)) model of loops .This formula was inspired by an attempt to understand the observations of Thapper on the enumeration of FPLs with prescribed connectivity, itself based on earlier work . In fact, we shall show in what follows that our new conjecture implies both the RS conjecture and the conjectures of .Since this article was written, the RS conjecture was proved in ;however there is no direct connection between our results and this proof, which provides no explicit formulae for the counting of FPLs.