Triangular fully packed loop configurations of excess 2

TL;DR

This paper derives a linear formula for counting triangular fully packed loop configurations with excess 2, building on previous results for lower excesses, and connects these counts to stable configurations invariant under Wieland drift.

Contribution

It introduces a linear expression for the number of TFPLs with excess 2, extending enumeration methods for configurations with lower excesses and relating them to stable TFPLs.

Findings

Provides a linear formula for TFPLs with excess 2

Connects counts to stable TFPLs invariant under Wieland drift

Extends enumeration results for TFPLs with lower excesses

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 $d(u)+d(v)\leq d(w)$, 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 $A_\pi$ of FPLs corresponding to a given link pattern $\pi$. 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 $d(w)-d(u)-d(v)=0,1$.