Special Integers - State Definition

Define $dp(pos, tight, started, mask)$ = count of valid special integers from position $pos$ onward. $mask$: bitmask of used digits. Bit $d$ is $1$ if digit $d$ has been placed.

Why "started"? Leading zeros don't count as using digit $0$. The number $042$ is $42$, and $0$ isn't "used." You only start tracking the mask once you place a real digit. Choosing what dp tracks is often the hardest part. Each dp value must have enough information to count valid numbers from that position onward.