We study the asymptotic behavior of the rank statistic for unimodal sequences. We use analytic techniques involving asymptotic expansions in order to prove asymptotic formulas for the moments of the rank. Furthermore, when appropriately normalized, the values of the unimodal rank asymptotically follow a logistic distribution. We also prove similar results for Durfee unimodal sequences and semi-strict unimodal sequences, with the only major difference being that the (normalized) rank for semistrict unimodal sequences has a distributional limit of a point mass probability distribution.

中文翻译：

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单峰序列秩的渐近分布

我们研究了单峰序列的秩统计的渐近行为。我们使用涉及渐近展开的分析技术来证明秩矩的渐近公式。此外，当适当归一化时，单峰秩的值渐近地遵循逻辑分布。我们还证明了 Durfee 单峰序列和半严格单峰序列的相似结果，唯一的主要区别是半严格单峰序列的（归一化）秩具有点质量概率分布的分布限制。