For how many integers having between 1 and 10 digits (base 10) are all of their digits when read from left to right monotonically increasing? In other words, every digit is less than or equal to all of those to its right. For example, 244467889 is one of them, and 0 is another, but there are more.

(In reply to

found the mapping by Ady TZIDON)

Ady:

It sounds like your 10 minute solution is the same as (or isomorphic to) my posted solution. It is very interesting to me that we have 4 different solution methods for this problem, and that none of them are having a computer just "count the ways". For me, at least, that is part of what makes math fun.

Apologies, by the way, for posting the solution before you posted, as a result of which I only credited Jer and Charlie for having a correct answer. If I had waited I could have credited you also.

Steve