Counting & Probability · Hard problem

A function $f$ from a $5$-element set $\{1,2,3,4,5\}$ to a $9$-element set $\{a_1,a_2,\dots,a_9\}$ (with the $a_i$ listed in a fixed order) is called increasing if $f(1)<f(2)<f(3)<f(4)<f(5)$ under that ordering (so $f$ never repeats a value either). How many increasing functions are there?