Boundary emptiness formation probabilities in the six-vertex model at $\mathrm{\Delta }=-\frac{1}{2}$

We define a new family of overlaps $C_{N,m}$ for the XXZ Hamiltonian on a periodic chain of length $N$. These are equal to the linear sums of the groundstate components, in the canonical basis, wherein $m$ consecutive spins are fixed to the state $\uparrow$. We define the boundary emptiness formation probabilities as the ratios $C_{N,m}\mathrm{/}{C}_{N,0}$ of these overlaps. In the associated six-vertex model, they correspond to correlation functions on a semi-infinite cylinder of perimeter $N$. At the combinatorial point $\Delta = -\frac12$, we obtain closed-form expressions in terms of simple products of ratios of integers.