Let $I$ be the ideal in $\displaystyle\,k\left[x_{1,1}, \ldots, x_{m,n}\right]$ generated by the $k\times k$ permanents of the matrix $$ \begin{bmatrix} x_{1,1} & \cdots & x_{1,n}\\ \vdots & \ddots & \vdots\\ x_{m,1} & \cdots & x_{m,n}\\ \end{bmatrix}. $$
What is a primary decomposition of $I$?
I have figured out that in the $2 \times n$ case, the primary components are the vanishing of the top row, the vanishing of the bottom row, and then one permanent vanishes and all other variables vanish. In total, this is ${n \choose 2}+2$ primary components.
Related: Primary Decomposition of $ 2\times 2$ Permanents of a $2\times 3$ Matrix