不同多项式系数的大西顿子集
Large Sidon Subsets and Pair-Sum Multiplicities of Distinct Multinomial Coefficients
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中文总结 AI 辅助
该研究确定不同多项式系数集的西顿子集最大基数的下界，证明其为Ω(n log n)，给出Mₙ与s(n)差值的下界，还得到1≤n≤16时s(n)的精确值。
中文摘要 AI 辅助
对于正整数n，令Mₙ为n!/(p₁!…pₜ!)的不同值构成的集合，其中(p₁,…,pₜ)遍历n的整数分拆。我们研究Mₙ的西顿子集的最大基数s(n)。集合Mₙ满足乘积嵌入，由此得到基本提升不等式s(n) ≥ s(r) + n - r。结合强西顿子集的三分之一抽取引理、西顿子集的一般结果以及无限制素数分拆，我们证明当n→∞时，liminf[log(s(n)/n)·log log n / √log n] ≥ π/√3，特别地，s(n) = Ω(n log n)。另一方面，我们在Mₙ中构造了多个两两不交的等差数列，令M(n)=|Mₙ|，则当n→∞时，liminf[(M(n)-s(n))/(n^(3/2)·√log n)] ≥ 2/(3√3)。我们还通过穷举SAT计算得到了1≤n≤16时s(n)的精确值，补充材料提供了完整的Maple源代码、n=14、15、16的控制台输出以及编码说明。
英文摘要
For a positive integer $n$, let $\mathcal{M}_n$ be the set of distinct multinomial coefficient values $n!/(p_1!\cdots p_t!)$, where $(p_1,\ldots,p_t)$ ranges over the integer partitions of $n$. We study two complementary aspects of the additive structure of $\mathcal{M}_n$: the maximum cardinality $s(n)$ of a Sidon subset and the multiplicities of unordered pair sums in the full set. A product embedding, strongly Sidon subsets, and estimates for prime partitions give \[ \liminf_{n\to\infty}\frac{\log(s(n)/n)\log\log n}{\sqrt{\log n}}\geq\fracπ{\sqrt{3}}. \] Long arithmetic progressions and separated copies yield a complementary lower bound $2/(3\sqrt{3})$ for the normalized Sidon defect. Writing $r_n(t)$ for the number of unordered representations $t=x+y$ with $x,y\in\mathcal{M}_n$, we study the collision excess $C(n)$, the number $D(n)$ of multiply represented sums, the maximum multiplicity $μ(n)$, and the cumulative profile $T(n,k)$. We prove \[ \liminf\frac{C(n)}{n^2\log n}\geq\frac18,\qquad \liminf\frac{D(n)}{n^{3/2}\sqrt{\log n}}\geq\frac{4}{3\sqrt{3}},\qquad \liminf\frac{μ(n)}{\sqrt{n\log n}}\geq\frac12. \] We also obtain a scaled lower envelope for the full profile, an additive-energy bound, stabilization with respect to the number of variables, and an explicit family of trinomial collisions. Exact Sidon values through $n=16$, the lower bound $s(17)\geq89$, and pair-sum statistics through $n=20$ are reported with reproducible computational material.
