We introduce the poset of mesh patterns, which generalises the permutation pattern poset. We fully classify the mesh patterns for which the interval [1^\emptyset,m] is non-pure, where 1^\emptyset is the unshaded singleton mesh pattern. We present some results on the M\"obius function of the poset, and show that {\mu}(1^\emptyset,m) is almost always zero. Finally, we introduce a class of disconnected and non-shellable intervals by generalising the direct product operation from permutations to mesh patterns.

中文翻译：

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网格模式的偏序

我们介绍了网格模式的poset，它概括了置换模式poset。我们对区间 [1^\emptyset,m] 为非纯的网格模式进行了全面分类，其中 1^\emptyset 是未着色的单例网格模式。我们给出了偏序集的 M\"obius 函数的一些结果，并表明 {\mu}(1^\emptyset,m) 几乎总是为零。最后，我们通过推广引入了一类断开的和不可壳的区间从排列到网格模式的直接乘积操作。