斜率簇的上下数与初始单项式
Updown numbers and the initial monomials of the slope variety
摘要：
设$I_n$是在平面上放置$n$点并用直线连接每对点所形成的$\binom{n}{2}$直线的斜率上的所有代数关系的理想。在两个自然项序下，$I_n$的初始理想由满足一定模式避免条件的置换所对应的单项式生成。我们双射表明，这些排列枚举的updown（或欧拉）的号码，从而获得一个公式的数量的生成器的初始理想的$I_n$在每个程度。
英文摘要：
Let $I_n$ be the ideal of all algebraic relations on the slopes of the $\binom{n}{2}$ lines formed by placing $n$ points in a plane and connecting each pair of points with a line. Under each of two natural term orders, the initial ideal of $I_n$ is generated by monomials corresponding to permutations satisfying a certain pattern-avoidance condition. We show bijectively that these permutations are enumerated by the updown (or Euler) numbers, thereby obtaining a formula for the number of generators of the initial ideal of $I_n$ in each degree.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
