Counting Tilings - Transition Formula

For each valid transition from $cur\_mask$ to $next\_mask$, add $dp[c][cur\_mask]$ to $dp[c+1][next\_mask]$. This sums all ways to reach the new profile. Precompute transitions: for each $cur\_mask$, run the recursive placement function to find all reachable $next\_mask$ values.

Store these in $trans[cur\_mask]$. Then the DP loop becomes: for each $c$, for each $cur\_mask$, for each $next\_mask$ in $trans[cur\_mask]$, update. This separation makes code cleaner and faster. You compute transitions once and reuse them for all columns.