Counting Tilings - State Definition

Let $dp[c][mask]$ = number of ways to tile columns $0$ to $c-1$ completely, with column $c$ having profile $mask$. The profile tells you which cells in column $c$ are already filled by vertical tiles from column $c-1$. Base case: $dp[0][0] = 1$.

You start with no columns filled and an empty profile. All other $dp[0][mask]$ are $0$ because you can't have a non-empty profile before placing any tiles. Goal: $dp[m][0]$. After filling all $m$ columns, the final profile must be empty (no tiles sticking out). This counts all complete tilings.