Triangular fully packed loop configurations of excess 2

2015/06/02 by Sabine Beil, Beil, Sabine
Mathematics · #Combinatorics (math.CO) #FOS: Mathematics #math.CO

paper · pdf · doi:10.48550/arxiv.1506.00943

arxiv created 2015/06/02 · arxiv updated 2015/06/03

Abstract

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 which must necessarily satisfy that d(u)+d(v)≤ 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 A_π of FPLs corresponding to a given link pattern π. Later, Wieland drift - a map on TFPLs that is based on Wieland gyration - was defined. The main contribution of this article is 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 is consistent with already existing enumeration results for TFPLs with boundary (u,v;w) where d(w)-d(u)-d(v)=0,1.

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