Digit Set - State Definition

Define $dp(pos, tight)$ = count of valid numbers using positions $pos$ to end, given whether we're tight. No "started" flag needed here because all digits in the set are non-zero (the problem guarantees digits are from $1$-$9$).

Every number we form is positive. The answer is the sum of: (1) all numbers shorter than $n$, plus (2) $dp(0, true)$ for numbers with the same length as $n$. This state captures all the information needed to count valid numbers from position pos onward.