Arithmetic properties of some 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 $$\mathrm{per}{\left[1-{\zeta }^j{x}_k\right]}_{1\le j,k\le n}\text{and}\mathrm{per}{\left[\frac{1}{1-{\zeta }^{j-k}x}\right]}_{1\le j,k\le n}.$$ Another typical result states that for any odd integer $n>1$ we have $$t_n:=\frac{1}{\sqrt{n}}\mathrm{per}{\left[\tan \pi \frac{jk}{n}\right]}_{1\le j,k\le (n-1)/2}\in \mathbb{Z},$$ and that $t_p\equiv (-1{)}^{(p+1)/2}(\mathrm{mod}p)$ for any odd prime $p$. We also pose several conjectures for further research; for example, we conjecture that $$\mathrm{per}[|j-k|{]}_{1\le j,k\le p}\equiv -\frac{1}{2}(\mathrm{mod}p)$$ for any odd prime $p$.

28 pages, polished version with typos corrected

References in corpus (2)

• On some determinants involving the tangent function
• On permutations of $\{1,\dots ,n\}$ and related topics