Counting Tilings - Filling a Column

To fill column $c$ with profile $cur\_mask$, process rows $0$ to $n-1$. If bit $i$ is set, that cell is already occupied (skip). Otherwise, you must fill it. Two choices: place a vertical tile (occupies current and next column, setting bit $i$ in $next\_mask$) or horizontal tile (occupies current column only, but row $i+1$ must also be empty and gets filled).

This recursive function tries all valid placements and accumulates ways to reach each $next\_mask$. Only when all rows are processed does $next\_mask$ become a valid next profile.