Telescopic Structure of Numerical Semigroups Generated by Unsigned Stirling Numbers of the First Kind

Abstract

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.