Exact enumeration of lozenge tilings of a triangular region

We prove that the number of lozenge tilings of a certain triangular region $\mathcal{T}_n$ is given by the formula \[T_n=\prod_{\substack{1\leq a<b\leq 3n+2\\(a,b)\not=(n+1,2n+2)}}\left|1+\zeta^a+\zeta^b\right|^{1/3},\] where $\zeta=e^{2\pi i/(3n+3)}$.

This answers a question of Ciucu and Krattenthaler, both by finding the exact formula and by explaining why $T_n$ has many prime factors.

The proof reduces the lozenge tiling enumeration problem to evaluating the determinant of the bipartite adjacency matrix $M_n$ of the dual graph of $\mathcal{T}_n$, and then evaluates this determinant by diagonalising $M_n$.