AI 中文总结
研究三角形区域\(\mathcal{T}_n\)菱形平铺数量，通过将问题转化为求对偶图二分邻接矩阵\(M_n\)的行列式并对角化来求解，给出精确公式并解释其有许多质因数的原因。
AI 中文摘要
我们证明了某三角形区域\(\mathcal{T}_n\)的菱形平铺数量由公式\[T_n=\prod_{\substack{1\leq a<b\leq 3n+2\\(a,b)\not=(n+1,2n+2)}}\left|1+\zeta^a+\zeta^b\right|^{1/3}\]给出，其中\(\zeta=e^{2\pi i/(3n+3)}\)。这回答了Ciucu和Krattenthaler的一个问题，既给出精确公式又解释了\(T_n\)有许多质因数的原因。证明将菱形平铺计数问题归结为求\(\mathcal{T}_n\)对偶图的二分邻接矩阵\(M_n\)的行列式，然后通过对角化\(M_n\)来计算此行列式。
英文摘要
We prove that the number of lozenge tilings of a certain triangular region $\mathcal{T}_n$ is given by the formula \[T_n=\prod_{\substack{1\leq a<b\leq 3n+2\\(a,b)\not=(n+1,2n+2)}}\left|1+ζ^a+ζ^b\right|^{1/3},\] where $ζ=e^{2πi/(3n+3)}$. 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 $M_n$.
Comments12 pages, 5 figures. Corrected error in the statement of Corollary 1.3
