Arithmetic properties of some permanents

TL;DR

This paper explores the arithmetic properties of certain permanents involving roots of unity and trigonometric functions, providing explicit formulas, integrality results, and conjectures related to prime moduli.

Contribution

It introduces new closed-form formulas for permanents with roots of unity and trigonometric entries, and establishes integrality and congruence properties, along with conjectures for future research.

Findings

Explicit formulas for permanents involving roots of unity.

Proves integrality of certain tangent-based permanents for odd n.

Establishes congruences modulo primes for specific permanents.

Abstract

In this paper we study arithmetic properties of some permanents, many of which involve trigonometric functions. For any primitive $n$-th root $\zeta$ of unity, we obtain closed formulas for the permanents $$\mathrm{per}\left[1-\zeta^jx_k\right]_{1\le j,k\le n}\ \ \text{and}\ \ \mathrm{per}\left[\frac1{1-\zeta^{j-k}x}\right]_{1\le j,k\le n}.$$ Another typical result states that for any odd integer $n>1$ we have $$t_n:=\frac1{\sqrt n}\mathrm{per}\left[\tan\pi\frac{jk}n\right]_{1\le j,k\le (n-1)/2}\in\mathbb Z,$$ and that $t_p\equiv(-1)^{(p+1)/2}\pmod p$ for any odd prime $p$. We also pose several conjectures for further research; for example, we conjecture that $$\mathrm{per}[|j-k|]_{1\le j,k\le p}\equiv-\frac12\pmod p$$ for any odd prime $p$.