Av(1324)增长速率的新下界
A new lower bound for the growth rate of Av(1324)
AI总结：
该研究针对长度4模式的最后未知Stanley-Wilf极限Av(1324)，改进下界至10.617，通过放宽交错规则、移除Harris不等式等方法，得到更精确的计数结果。
AI中文摘要：
Av(1324)的增长速率是长度为4的模式对应的最后一个未知Stanley-Wilf极限。自Bevan、Brignall、Elvey Price和Pantone在2020年得到10.271012以来，最佳严格下界一直是该值；我们将其提高到10.617。他们的方案仅在一个方向上放宽了交错规则，而在两个方向上放宽该规则是有效的，此时Harris不等式会将所得计数下界约束为其两个边际的乘积。我们移除了该方案包含的最后一个不等式：连接单元的两个相邻点都放置在同一组斜分量序列上，因此它们的联合计数是单元状态空间平方上的单个转移算子，Catalan级数应用于的矩阵是幂幺的，因此该级数会终止且计数是精确的。其速率在条带轮廓中是凹的，这将最小化问题简化为有限多个顶点，并且对聚合权重未到达的顶点进行了分类。还引入了另外两个要素：多米诺集合相对于叶和空条带密度的代数倾斜（它们的构造在这些密度下成立），以及k叶条带密度的闭式，即使对于k=1，他们也未能得到该闭式。
英文摘要：
The growth rate of Av(1324) is the last unknown Stanley-Wilf limit of a length-four pattern. Since 2020 the best rigorous lower bound has been the 10.271012 of Bevan, Brignall, Elvey Price and Pantone; we raise it to 10.412263. Their construction lays 1324-avoiders along an infinite staircase of two-cell blocks alternating with single connecting cells, and forbids a 1324 by a local condition on how the points of a block interleave with the components of the cell beside it. Certain points of a block, its leaves, may be exempted from that condition. They exempt them on one axis only, and report that they could not count the possibilities when both are exempted. This paper exempts them on both axes and counts the result. The second relaxation is their own argument read in the other axis, and the Harris correlation inequality bounds the joint count below by the product of the two counts taken separately, at the cost of one factor of the component generating function.
