Combinatorics of hexagonal fully packed loop configurations

Abstract

In this article, fully packed loop configurations of hexagonal shape (HFPLs) are defined. They generalize triangular fully packed loop configurations. To encode the boundary conditions of an HFPL, a sextuple $(\mathsf{l}_\mathsf{T},\mathsf{t},\mathsf{r}_\mathsf{T};\mathsf{r}_\mathsf{B},\mathsf{b},\mathsf{l}_\mathsf{B})$ of $01$-words is assigned to it. In the first main result of this article, necessary conditions for the boundary $(\mathsf{l}_\mathsf{T},\mathsf{t},\mathsf{r}_\mathsf{T};\mathsf{r}_\mathsf{B},\mathsf{b},\mathsf{l}_\mathsf{B})$ of an HFPL are stated. For instance, the inequality $d(\mathsf{r}_\mathsf{B})+d(\mathsf{b})+d(\mathsf{l}_\mathsf{B})\geq d(\mathsf{l}_\mathsf{T})+d(\mathsf{t})+d(\mathsf{r}_\mathsf{T})+\vert\mathsf{l}_\mathsf{T}\vert_1\vert\mathsf{t}\vert_0+\vert\mathsf{t}\vert_1 \vert\mathsf{r}_\mathsf{T}\vert_0+\vert\mathsf{r}_\mathsf{B}\vert_0\vert\mathsf{l}_\mathsf{B}\vert_1$ has to be fulfilled, where $\vert\cdot\vert_i$ denotes the number of occurrences of $i$ for $i=0,1$ and $d(\cdot)$ denotes the number of inversions. The other main contribution of this article is the enumeration of HFPLs with boundary $(\mathsf{l}_\mathsf{T},\mathsf{t},\mathsf{r}_\mathsf{T};\mathsf{r}_\mathsf{B},\mathsf{b},\mathsf{l}_\mathsf{B})$ such that $d(\mathsf{r}_\mathsf{B})+d(\mathsf{b})+d(\mathsf{l}_\mathsf{B})-d(\mathsf{l}_\mathsf{T})-d(\mathsf{t})-d(\mathsf{r}_\mathsf{T})-\vert\mathsf{l}_\mathsf{T}\vert_1\vert\mathsf{t}\vert_0- \vert\mathsf{t}\vert_1\vert\mathsf{r}_\mathsf{T}\vert_0-\vert\mathsf{r}_\mathsf{B}\vert_0\vert\mathsf{l}_\mathsf{B}\vert_1=0,1$. To be more precise, in the first case they are enumerated by Littlewood-Richardson coefficients and in the second case their number is expressed in terms of Littlewood-Richardson coefficients.