The rank of the $5\times 5$ permanent tensor is sixteen

In this paper, we show that the tensor rank of the $5\times 5$ permanent tensor is at least $16$ over $\mathbb{C}$ and its subfields, which matches the known upper bound. The $5\times 5$ permanent tensor is a symmetric tensor corresponding to the monomial $x_1{x}_2{x}_3{x}_4{x}_5$ whose Waring rank is known to be $16$. Previously, it was known that its tensor rank is either $15$ or $16$ by the higher-order Koszul flattening and Glynn’s formula.