Bouncy Numbers
Working from left-to-right if no digit is exceeded by the digit to its left it is called an increasing number; for example, $134468$.
Similarly if no digit is exceeded by the digit to its right it is called a decreasing number; for example, $66420$.
We shall call a positive integer that is neither increasing nor decreasing a "bouncy" number; for example, $155349$.
Clearly there cannot be any bouncy numbers below one-hundred, but just over half of the numbers below one-thousand ($525$) are bouncy. In fact, the least number for which the proportion of bouncy numbers first reaches $50\%$ is $538$.
Surprisingly, bouncy numbers become more and more common and by the time we reach $21780$ the proportion of bouncy numbers is equal to $90\%$.
Find the least number for which the proportion of bouncy numbers is exactly $99\%$.