Can you decide if there is a coincidence in the numbers counting two different combinatorial objects? For example, can you decide if two regions in $\mathbb{R}^3$ have the same number of domino tilings? There are two versions of the problem, with $2\times 1 \times 1$ and $2\times 2 \times 1$ boxes. We prove that in both cases the coincidence problem is not in the polynomial hierarchy unless the polynomial hierarchy collapses to a finite level. While the conclusions are the same, the proofs are notably different and generalize in different directions. We proceed to explore the coincidence problem for counting independent sets and matchings in graphs, matroid bases, order ideals and linear extensions in posets, permutation patterns, and the Kronecker coefficients. We also make a number of conjectures for counting other combinatorial objects such as plane triangulations, contingency tables, standard Young tableaux, reduced factorizations and the Littlewood--Richardson coefficients.

翻译：你能判断两个不同组合对象计数结果是否相同吗？例如，能否判断$\mathbb{R}^3$中两个区域的多米诺骨牌铺砌数量是否相等？该问题存在两个版本，分别涉及$2\times 1 \times 1$和$2\times 2 \times 1$的盒子。我们证明，在这两种情况下，除非多项式层级坍缩至有限层，否则巧合问题不属于该层级。尽管结论相同，但证明方法显著不同且在不同方向上具有推广性。我们进而探讨了以下对象的计数巧合问题：图中的独立集与匹配、拟阵基、偏序集中的理想与线性扩展、排列模式以及Kronecker系数。此外，我们还对平面三角剖分、列联表、标准杨表、简化因子分解及Littlewood-Richardson系数等其他组合对象的计数提出了一系列猜想。