I don't know much about Schubert calculus, but I would like to know all of the intersection numbers $$ \#(X_{w_1} \cap X_{w_2} \cap X_{w_3}) $$ where $X_w$ indicates a Schubert variety (maybe translated) in $\mathrm{GL}_5/B$, $w_1, w_2, w_3$ are in $S_5$ and $w_1 w_2 w_3$ is the long element of $S_5$.

There are a priori too many such numbers to fit in a table (medium-sized fraction of 120^3), but maybe there are some symmetries and redundancies that cut it down?

Failing that, is there some computer code that will tell me all of these numbers?