設 n 和 f 為兩個正整數。在本文中，我們將研究對於 n 之整數分割，其任何子集總和皆不等於 f 的最大基數分割 P 的特性。Tan 等人 [3] 發現，若固定 f ，當 n 超過一定的閾值 τf 時，P 中的部分數會隨著 n 的增加表現出一種循環特性。具體來說，他們證明了 τf = O(f L2) ，其中 L 表示 f 的最小正非除數。也就是說，對於任意 f ，循環性質總是存在的。
Tan 等人猜想 τf = Θ(f L) 。本文證明了此猜想的正確性。

Let n and f be two positive integers. In this thesis, we investigate the maximal cardinality integer partition P of n which does not contain any subset P ′ ⊆ P with sum equal to f . Tan et al. [3] showed that when f is fixed, the number of parts in P exhibits a cyclic property, as n increases and n exceeds a certain threshold τf . In particular, they showed that τf = O(f L2), where L denotes the smallest positive non-divisor of f . That is, the cyclic property always occurs for any f .
It was conjectured that τf = Θ(f L). This thesis shows that this conjecture is true.

Contents
Abstract (Chinese) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . I
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . II
Acknowledgments (Chinese) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . III
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IV
Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VII
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VIII
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 Basic Definitions and Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
3 Upper Bound of the Cyclic Property . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3.1 Minimum Extensible by ν > f . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3.2 Minimum Extensible by ν ∈ (L, f ) . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.3 Deriving the Upper Bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.3.1 When f < 2L(L - 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.3.2 Connection between “minimum extensible by L” and cyclic property . . . . . . . . 21
4 Lower Bound of the Cyclic Property . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4.1 When f is odd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
4.2 When f is even . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

[1] G. H. Hardy and S. Ramanujan. “Asymptotic Formulaae in Combinatory Analysis”. In: Proceedings of the London Mathematical Society 2-17.1 (1918), pp. 75–115. doi: 10.1112/plms/s2-17.1.75.
[2] Tanya Khovanova and Konstantin Knop. Coins of Three Different Weights. 2014. arXiv: 1409.0250 [math.HO].
[3] Te-Sheng Tan, Dai-Yang Wu, and Wing-Kai Hon. “Partitions of n that avoid partitions of f, and an application to the tiny-pan coin weighing problem”. In: Discrete Mathematics 340.6 (2017), pp. 1397–1404. issn: 0012-365X. doi: https://doi.org/10.1016/j.disc.2016.09.034.
[4] Eric W Weisstein. “Ferrers diagram”. From MathWorld–A Wolfram Web Resource. [Accessed online on June 20, 2023.] url: https : / / mathworld . wolfram.com/FerrersDiagram.html.