We study the gaps Δ p n between consecutive rank sizes in r -differential posets by introducing a projection operator whose matrix entries can be expressed in terms of the number of certain paths in the Hasse diagram. We strengthen Miller’s result that Δ p n ≥ 1, which resolved a longstanding conjecture of Stanley, by showing that Δ p n ≥ 2 r . We also obtain stronger bounds in the case that the poset has many substructures called threads .

中文翻译：

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差分姿态中的路径计数和等级差距

我们通过引入投影算子来研究 r 差分偏序组中连续秩大小之间的差距 Δ pn ，其矩阵条目可以根据哈斯图中某些路径的数量来表示。我们通过证明 Δ pn ≥ 2 r 加强了米勒的结果，即 Δ pn ≥ 1，该结果解决了斯坦利长期以来的猜想。在poset有许多称为线程的子结构的情况下，我们也获得了更强的边界。