The \emph{Stanley--Yan} (SY) \emph{inequality} gives the ultra-log-concavity for the numbers of bases of a matroid which have given sizes of intersections with $k$ fixed disjoint sets. The inequality was proved by Stanley (1981) for regular matroids, and by Yan (2023) in full generality. In the original paper, Stanley asked for equality conditions of the SY~inequality, and proved total equality conditions for regular matroids in the case $k=0$. In this paper, we completely resolve Stanley's problem. First, we obtain an explicit description of the equality cases of the SY inequality for $k=0$, extending Stanley's results to general matroids and removing the ``total equality'' assumption. Second, for $k\ge 1$, we prove that the equality cases of the SY inequality cannot be described in a sense that they are not in the polynomial hierarchy unless the polynomial hierarchy collapses to a finite level.

翻译：\emph{斯坦利-严} (SY) \emph{不等式}给出了拟阵基数的超对数凹性，这些基数与$k$个固定不相交集合具有给定大小的交集。该不等式由Stanley (1981) 对正则拟阵证明，并由Yan (2023) 在完全一般性下证明。在原始论文中，Stanley询问了SY不等式的等式条件，并在$k=0$的情形下证明了正则拟阵的完全等式条件。在本文中，我们完全解决了Stanley的问题。首先，我们获得了$k=0$时SY不等式等式情形的明确描述，将Stanley的结果推广到一般拟阵并移除了"完全等式"的假设。其次，对于$k\ge 1$，我们证明了SY不等式的等式情形在某种意义上无法被描述，即它们不在多项式谱系中，除非多项式谱系坍缩到有限层级。