The open XXZ chain at $\mathrm{\Delta }=-1\mathrm{/}2$ and totally-symmetric alternating sign matrices

The open XXZ spin chain with the anisotropy parameter $\mathrm{\Delta }=-\frac{1}{2}$, diagonal boundary fields that depend on a parameter $x$, and finite length $N$ is studied. In a natural normalisation, the components of its ground-state vector are polynomials in $x$ with integer coefficients. It is shown that their sum is given by a generating function for the weighted enumeration of totally-symmetric alternating sign matrices with weights depending on $x$.