The Sharp Exponential Constant for 6-Regular Circulant Matrices
2026-09-24
Research publication
The Sharp Exponential Constant for 6-Regular Circulant Matrices
Preprint
- Original date
- Authors
- Yair Lavi
- Combinatorics
- Permanents
- Circulant matrices
- Matching entropy
Abstract
We prove that, for every integer \(n\ge6\) and six-element set \(S\subseteq\mathbb Z_n\), the binary circulant \(C_n(S)=(\mathrm{1}_{\{j-i\in S\}})_{i,j\in\mathbb Z_n}\) satisfies
\[ \operatorname{per}C_n(S)\ge\left(\frac{3125}{1296}e^{7/250000}\right)^n. \]
For every integer \(d\ge2\), we also identify the optimal uniform base as the limit of an explicit sequence of circulants:
\[ \inf_{n\ge d}\ \inf_{\substack{S\subseteq\mathbb Z_n\\|S|=d}}\operatorname{per}C_n(S)^{1/n}=\lim_{q\to\infty}\operatorname{per}C_{q^{d-1}}(\{0,1,q,\ldots,q^{d-2}\})^{1/q^{d-1}}, \]
where \(q\) runs through integers at least \(2\). In degree six this base is at most \(163812873663198^{1/35}\). The lower bound strictly improves the sharp uniform base \(3125/1296\) for unrestricted binary \(6\)-regular matrices. This resolves the surviving circulant lower-bound question in Minc’s Problem 10; its proposed full upper bound fails even for circulants.