AMC 12 2025a Problem 23

Call a positive integer fair if no digit is used more than once, it has no $0$s, and no digit is adjacent to two greater digits. For example, $196, 23,$ and $12463$ are fair, but $1546, 320$ and $34321$ are not. How many fair positive integers are there?

$\textbf{(A) } 511 \textbf{(B) } 2584 \textbf{(C) } 9841 \textbf{(D) } 17711 \textbf{(E) } 19682$