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

Let $$S_n=⟨\left[\genfrac{}{}{0px}{}{n}{1}\right],\dots ,\left[\genfrac{}{}{0px}{}{n}{n-1}\right]⟩$$ be the numerical semigroup generated by the nontrivial unsigned Stirling numbers of the first kind in the $n$th row. Writing $a=\left[\genfrac{}{}{0px}{}{n}{n-1}\right]=\left(\genfrac{}{}{0px}{}{n}{2}\right)$ and $b_j=\left[\genfrac{}{}{0px}{}{n}{n-2j}\right]$, we determine the complete gcd filtration of the reduced generating system. More precisely, if $$M_j=\frac{1}{2}\mathrm{lcm}\{m\ge 1:\phi (m)\le 2j\},$$ then $$\gcd (a,b_1,\dots ,b_j)=\frac{a}{\gcd (a,M_j)}\mathrm{.}$$ 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.