AI 中文总结
本文研究对称群特征标表中无零列的数量$D(n)$，改进其渐近上界，对几乎所有$n$给出更紧界，提出相关猜想并完成形式化验证。
AI 中文摘要
对称群$S_n$的特征标表的行与列均自然由$n$的分拆索引。设$D(n)$表示$S_n$中列不含零元的共轭类数量，恒等列始终无零元，故$D(n)\geq1$。已知$D(n)\ll n^2$，本文证明$D(n)\ll n^{3/4}$；其次，利用Matomäki与Radziwill的工作，对几乎所有正整数$n$，证明对任意$B>5/6$，$D(n)\ll_B n^{1/2}(\log n)^B$，并给出例外集的定量界；最后，提出支持$D(n)\ll_{\varepsilon} n^{\varepsilon}$猜想的启发式论证。AxiomProver在Lean中结合已有文献对本文结果进行了形式化验证。
英文摘要
The rows and columns of the character table of the symmetric group $S_n$ are both naturally indexed by partitions of $n$. Let $D(n)$ denote the number of conjugacy classes of $S_n$ whose column contains no zero entry. The identity column is always zero-free, so $D(n)\geq 1$. It is known that $D(n)\ll n^2$. We prove that $D(n)\ll n^{3/4}$. Second, we prove for almost all positive integers $n$ that $D(n)\ll_B n^{1/2}(\log n)^B$ for every $B>5/6$, with a quantitative bound for the exceptional set, using work of Matomäki and Radziwill. Finally, we offer a heuristic supporting our conjecture that $D(n)\ll_{\varepsilon} n^{\varepsilon}$. AxiomProver formalized the results in this paper in Lean assuming preexisting literature.
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