论文标题
等级的单程和斯坦顿的猜想证明
Unimodality of ranks and a proof of Stanton's conjecture
论文作者
论文摘要
最近，对分区功能之间的各种不平等广泛的关注。例如，尼古拉斯（Nicolas）{and and desavlvo-pak，}证明了$ p（n）$最终是log-concave，而ji-zang表明曲柄最终是单峰。这导致了一系列最近在不同方向上推广此类结果的活动。同时，斯坦顿最近对与分区等级和曲柄相关的某些多项式的积极性做出了深刻的猜想，其最终目标是指向``更深''结构的精制等级和曲柄的道路。这些在最近的作品中已被证明是强大的，在彩色分区的情况下，这些猜想已经确定了此类猜想的进一步无限家族。在本文中，我们采用圆形方法来证明等级的非模样。作为推论，我们证明了斯坦顿的原始猜想。这表明了未来对Stanton猜想存在的正积分系数的研究，这暗示了尚未发现的新组合结构。
Recently, much attention has been given to various inequalities among partition functions. For example, Nicolas, {and later DeSavlvo--Pak,} proved that $p(n)$ is eventually log-concave, and Ji--Zang showed that the cranks are eventually unimodal. This has led to a flurry of recent activity generalizing such results in different directions. At the same time, Stanton recently made deep conjectures on the positivity of certain polynomials associated to ranks and cranks of partitions, with the ultimate goal of pointing the way to ``deeper'' structure refining ranks and cranks. These have been shown to be robust in recent works, which have identified further infinite families of such conjectures in the case of colored partitions. In this paper, we employ the Circle Method to prove unimodality for ranks. As a corollary, we prove Stanton's original conjecture. This points to future study of the positive, integral coefficients Stanton conjectured to exist, hinting at new combinatorial structure yet to be uncovered.
