Antichains among Divisor Sums of Divisors

Abstract

For a positive integer n, let S(n)={σ(d): d| n} be ordered by divisibility, and let a(n) be its width. We study how much of the divisor lattice of n survives under the map d↦ σ(d). For every fixed prime-exponent pattern, the largest possible value of a(n) is the width of a corresponding product of divisibility posets, and this maximum is attained infinitely often. In the squarefree case with ω(n)=r, every width from $1$ to binomr⌊ r/2⌋ occurs, again infinitely often. For repeated exponent patterns, Engel's limit theory reduces the asymptotic width problem to optimal representation variance. We show that for the divisibility poset D_m=({1,…,m},|), the prime directions $2$ and $3$ already determine the exact variance. The structural reason is the critical dilation $4$, lying strictly between $3$ and the next prime $5$. This yields an exact finite formula, the limit V(D_m)→ 1.0367934201598…, and asymptotics for repeated exponent patterns. The complementary minimum-width problem remains largely open.

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BibTeX RIS

BibTeX

@misc{huber2026antichains,
  title = {{Antichains among Divisor Sums of Divisors}},
  author = {Felix Huber},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.02264},
  eprint = {2610.02264},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.02264},
}

RIS

TY  - UNPB
TI  - Antichains among Divisor Sums of Divisors
AU  - Felix Huber
PY  - 2026
DA  - 2026/10/01
PB  - arXiv
DO  - 10.48550/arxiv.2610.02264
UR  - https://arxiv.org/abs/2610.02264
AB  - For a positive integer n, let S(n)={σ(d): d| n} be ordered by divisibility, and let a(n) be its width. We study how much of the divisor lattice of n survives under the map d↦ σ(d). For every fixed prime-exponent pattern, the largest possible value of a(n) is the width of a corresponding product of divisibility posets, and this maximum is attained infinitely often. In the squarefree case with ω(n)=r, every width from $1$ to binomr⌊ r/2⌋ occurs, again infinitely often. For repeated exponent patterns, Engel's limit theory reduces the asymptotic width problem to optimal representation variance. We show that for the divisibility poset D_m=({1,…,m},|), the prime directions $2$ and $3$ already determine the exact variance. The structural reason is the critical dilation $4$, lying strictly between $3$ and the next prime $5$. This yields an exact finite formula, the limit V(D_m)→ 1.0367934201598…, and asymptotics for repeated exponent patterns. The complementary minimum-width problem remains largely open.
ER  -

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