Telescopic Structure of Numerical Semigroups Generated by Unsigned Stirling Numbers of the First Kind
Abstract
Let Sn=⟨ genfrac[]0ptn1,…,genfrac[]0ptnn−1⟩ be the numerical semigroup generated by the nontrivial unsigned Stirling numbers of the first kind in the nth row. Writing a=genfrac[]0ptnn−1=binom n2 and bj=genfrac[]0ptnn−2j, we determine the complete gcd filtration of the reduced generating system. More precisely, if Mj=1/2lcm{m≥1:φ(m)≤2j}, then gcd(a,b1,…,bj)=a/gcd(a,Mj). 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 Sn is rectangular, yielding an explicit Frobenius formula. The same structure shows that Sn is free and symmetric and gives explicit formulas for its genus and conductor, while its type is one.
Related
- The structure of almost symmetric almost complete intersection numerical semigroups
- Powers in orbits and semigroups generated by polynomials
- Weak mixing and non-recurrence along finitely generated multiplicative semigroups of ℕ
- Irreducibility of Endomorphisms of Finitely Generated Free Semigroups
- The type and cardinality of minimal presentations of numerical semigroups with embedding dimension four
BibTeX
@misc{komatsu2026telescopic,
title = {{Telescopic Structure of Numerical Semigroups Generated by Unsigned Stirling Numbers of the First Kind}},
author = {Takao Komatsu and Kyunghwan Song},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.03070},
eprint = {2610.03070},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.03070},
}
RIS
TY - UNPB
TI - Telescopic Structure of Numerical Semigroups Generated by Unsigned Stirling Numbers of the First Kind
AU - Takao Komatsu
AU - Kyunghwan Song
PY - 2026
DA - 2026/10/02
PB - arXiv
DO - 10.48550/arxiv.2610.03070
UR - https://arxiv.org/abs/2610.03070
AB - Let S_n=⟨ genfrac[]0ptn1,…,genfrac[]0ptnn−1⟩ be the numerical semigroup generated by the nontrivial unsigned Stirling numbers of the first kind in the nth row. Writing a=genfrac[]0ptnn−1=binom n2 and b_j=genfrac[]0ptnn−2j, we determine the complete gcd filtration of the reduced generating system. More precisely, if M_j=1/2lcm{m≥1:φ(m)≤2j}, then gcd(a,b_1,…,b_j)=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.
ER -