The principal specialization ${\nu }_w={\mathfrak{S}}_w(1,\dots ,1)$ of the Schubert polynomial at $w$, which equals the degree of the matrix Schubert variety corresponding to $w$, has attracted a lot of attention in recent years. In this paper, we show that ${\nu }_w$ is bounded below by $1+p_{132}\left(w\right)+p_{1432}\left(w\right)$ where $p_u\left(w\right)$ is the number of occurrences of the pattern $u$ in $w$, strengthening a previous result by A. Weigandt. We then make a conjecture relating the principal specialization of Schubert polynomials to pattern containment. Finally, we characterize permutations $w$ whose RC-graphs are connected by simple ladder moves via pattern avoidance.

中文翻译：

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Schubert多项式的主要专业和模式包含

主要专业 ${\nu }_w={\mathfrak{小号}}_w（\mathrm{1个}，\dots ，\mathrm{1个}）$ 的舒伯特多项式 $w$，等于矩阵Schubert的度数对应于 $w$，近年来引起了很多关注。在本文中，我们表明${\nu }_w$ 受以下限制 $\mathrm{1个}+p_{132}（w）+p_{1432}（w）$ 哪里 $p_{ü}（w）$ 是模式的出现次数 $ü$ 在 $w$，加强了A. Weigandt的先前结果。然后我们进行一个猜想，将Schubert多项式的主要专业性与模式包含性联系起来。最后，我们描述排列$w$ 通过模式回避，通过简单的梯形运动将其RC图连接起来。