Order-ideal board representation ($B=\{(x,y):0\le y<h(x)\}$)

A downward-closed subset of a finite product grid can be represented by nonincreasing column heights. Occupancy is $0\le y<h(x)$; the condition $h(0)\ge \cdots \ge h(N-1)$ is exactly closure toward smaller coordinates. Removing the northeast rectangle anchored at a legal $(x,y)$ truncates each height to its right to $\min (h(i),y)$, preserving closure. This represents a potentially quadratic number of occupied cells using a linear number of boundary values.