发表机构
Institute of Science Tokyo; Jeju National University（东京科学大学; 济州国立大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究由第一类无符号斯特林数生成的数值半群，通过gcd滤链和因子支撑性质证明其约化生成序列为望远镜结构，从而得到Apéry集矩形、显式Frobenius公式及自由对称性。
AI 中文摘要
设 $$ S_n=\left\langle \genfrac{[}{]}{0pt}{}{n}{1},\ldots,\genfrac{[}{]}{0pt}{}{n}{n-1}\right\rangle $$ 为第 $n$ 行非平凡第一类无符号斯特林数生成的数值半群。记 $a=\genfrac{[}{]}{0pt}{}{n}{n-1}=\binom n2$ 且 $b_j=\genfrac{[}{]}{0pt}{}{n}{n-2j}$，我们确定了约化生成系的完全 gcd 滤链。更精确地，若 $$ M_j=\frac12\operatorname{lcm}\{m\ge1:\varphi(m)\le2j\}, $$ 则 $$ \gcd(a,b_1,\ldots,b_j)=\frac{a}{\gcd(a,M_j)}. $$ 证明使用了素数幂块多项式以及阈值次数处的精确 $p$-adic 支撑。随后我们建立了典范混合基数约化的因子支撑性质，并将其与初等对称增长估计相结合，证明在移除不活跃生成元后，所得生成序列是望远镜的。因此，$S_n$ 的 Apéry 集是矩形的，从而得到显式的 Frobenius 公式。同一结构还表明 $S_n$ 是自由且对称的，并给出了其亏格和导子的显式公式，而其类型为 1。
英文摘要
Let $$ S_n=\left\langle \genfrac{[}{]}{0pt}{}{n}{1},\ldots,\genfrac{[}{]}{0pt}{}{n}{n-1}\right\rangle $$ be the numerical semigroup generated by the nontrivial unsigned Stirling numbers of the first kind in the $n$th row. Writing $a=\genfrac{[}{]}{0pt}{}{n}{n-1}=\binom n2$ and $b_j=\genfrac{[}{]}{0pt}{}{n}{n-2j}$, we determine the complete gcd filtration of the reduced generating system. More precisely, if \[ M_j=\frac12\operatorname{lcm}\{m\ge1:φ(m)\le2j\}, \] then $$ \gcd(a,b_1,\ldots,b_j)=\frac{a}{\gcd(a,M_j)}. $$ The proof uses prime-power block polynomials and exact $p$-adic support at threshold degrees. We then establish a divisor-support property for canonical mixed-radix reductions and combine it with an elementary-symmetric growth estimate to prove that, after inactive generators are removed, the resulting generating sequence is telescopic. Consequently, the Apéry set of $S_n$ is rectangular, yielding an explicit Frobenius formula. The same structure shows that $S_n$ is free and symmetric and gives explicit formulas for its genus and conductor, while its type is one.
Comments20 pages
