Suppose I have 9 integers such that
$i_1>i_2>i_3$
$i_4>i_5>i_6$
$i_7>i_1$
$i_2>i_4>i_8$
and $i_9>i_3>i_5$.
From this we then know that as $i_3>i_5$ and $i_4>i_5$ then $i_7>i_1>i_2>i_5$, and we can make similar deductions.
Given this information how many possible rankings of largest to smallest are there for $i_1$ through $i_9$?