Triangular Fully Packed Loop Configurations of Excess 2

Triangular fully packed loop configurations (TFPLs) came up in the study of fully packed loop configurations on a square (FPLs) corresponding to link patterns with a large number of nested arches. To a TFPL is assigned a triple (u, v;w) of 01-words encoding its boundary conditions that must necessarily satisfy d(u) + d(v) 6 d(w), where d(u) denotes the number of inversions in u. Wieland gyration, on the other hand, was invented to show the rotational invariance of the numbers of FPLs having given link patterns. Later, Wieland drift – a map on TFPLs that is based on Wieland gyration – was defined. The main contribution of this article will be a linear expression for the number of TFPLs with boundary (u, v;w) where d(w)− d(u)− d(v) = 2 in terms of numbers of stable TFPLs, that is, TFPLs invariant under Wieland drift. This linear expression generalises already existing enumeration results for TFPLs with boundary (u, v;w) that satisfies d(w)− d(u)− d(v) ∈ {0, 1}.