曲柄函数的无限族、Stanton型拓扑和单峰性
Infinite families of crank functions, Stanton-type conjectures, and unimodality
摘要：
戴森的秩函数和安德鲁斯-加文曲柄函数著名的组合证人拉马努金的分拆函数同余模5，7和11。虽然这些函数可以用来显示对应的分区集合被分成5、7或11个大小相等的集合，但人们可能会问，如何使由秩或曲柄组合显式组织的分区之间的结果双射。斯坦顿最近做了一些尝试，旨在揭示沿着这些路线的更深层次的组合结构，事实证明，需要对秩和曲柄进行微小的修改。在这里，我们证明了其中的两个命题。基于单峰性和对称性，我们还给出了某些分圆多项式的多项式导数具有非负系数的抽象判据。此外，我们将Stanton猜想推广到曲柄的无限族。这表明进一步应用到其他组合对象。我们还讨论了数值证据，我们的结构，连接与其他分析结构，如分配的分区排名。
英文摘要：
Dyson's rank function and the Andrews--Garvan crank function famously give combinatorial witnesses for Ramanujan's partition function congruences modulo 5, 7, and 11. While these functions can be used to show that the corresponding sets of partitions split into 5, 7, or 11 equally sized sets, one may ask how to make the resulting bijections between partitions organized by rank or crank combinatorially explicit. Stanton recently made conjectures which aim to uncover a deeper combinatorial structure along these lines, where it turns out that minor modifications of the rank and crank are required. Here, we prove two of these conjectures. We also provide abstract criteria for quotients of polynomials by certain cyclotomic polynomials to have non-negative coefficients based on unimodality and symmetry. Furthermore, we extend Stanton's conjecture to an infinite family of cranks. This suggests further applications to other combinatorial objects. We also discuss numerical evidence for our conjectures, connections with other analytic conjectures such as the distribution of partition ranks.
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