Not a Real World Problem

Consider a grid with $H$ rows (numbered $1$ through $H$) and $W$ columns (numbered $1$ through $W$). Let's denote the value in a cell in row $i$ and column $j$ by $h_{i,j}$. $N$ particles (numbered $1$ through $N$) are distributed in the cells in such a way that there is at most one particle in each cell. For each valid $i$, the $i$-th particle has a value $p_i$ and it is placed in the cell in row