Bounds for d-distinct partitions
Bounds for d-distinct partitions
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DOI：
10.46298/hrj.2021.7430
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发表时间：
2021-05
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通讯作者：
Soon-Yi Kang;Young Kim
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文献类型：
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作者：
Soon-Yi Kang;Young Kim
欧拉恒等式和罗杰斯-拉马努金恒等式可能是划分理论中最著名的结果。根据它们，n的1-相异分拆和2-相异分拆分别与n的±1模4全等分拆和n的±1模5全等分拆相等。此外，它们的生成函数是模函数，直到乘以q的有理数幂。然而，当d ≥ 3时，d-不同划分既没有相同类型的划分恒等式，也没有相同类型的模块性。相反，存在与d-不同划分相关的划分不等式和模拟模块性。例如，阿尔德-安德鲁斯定理指出n的d-不同划分的数量大于或等于n分成全等于±1（mod d+3）的部分的数量。本文介绍了Alder-Andrews定理的推广和类似的最新进展，并建立了d-不同划分的渐近上下界.利用渐近关系和计算得到的数据，我们提出了一个关于划分不等式的猜想，该不等式给出了d-不同划分的上界。具体地说，当d ≥ 4时，n的d-不同划分的数目小于或等于n划分成全等于±1（mod m）的部分的数目，其中m ≤ 2dπ^2 / [3 log^2（d）+6 log d]。
International audience Euler's identity and the Rogers-Ramanujan identities are perhaps the most famous results in the theory of partitions. According to them, 1-distinct and 2-distinct partitions of n are equinumerous with partitions of n into parts congruent to ±1 modulo 4 and partitions of n into parts congruent to ±1 modulo 5, respectively. Furthermore, their generating functions are modular functions up to multiplication by rational powers of q. For d ≥ 3, however, there is neither the same type of partition identity nor modularity for d-distinct partitions. Instead, there are partition inequalities and mock modularity related with d-distinct partitions. For example, the Alder-Andrews Theorem states that the number of d-distinct partitions of n is greater than or equal to the number of partitions of n into parts which are congruent to ±1 (mod d+3). In this note, we present the recent developments of generalizations and analogs of the Alder-Andrews Theorem and establish asymptotic lower and upper bounds for the d-distinct partitions. Using the asymptotic relations and data obtained from computation, we propose a conjecture on a partition inequality that gives an upper bound for d-distinct partitions. Specifically, for d ≥ 4, the number of d-distinct partitions of n is less than or equal to the number of partitions of n into parts congruent to ±1 (mod m), where m ≤ 2dπ^2 / [3 log^2 (d)+6 log d] .