The code here for \Pi_1 interprets each constraint (1,2,3) as "1 and 2 must both be
earlier in the permutation than 3", so ...1...2...3... and ...2...1...3.... both satisfy
this constraint. However, in the paper \Pi_1 is defined as "{ (123), (132) }". This
is isomorphic to the model used in the code, but is a different symmetry: here we require
1 to be earlier in the permutation than 2 and 3.

This is no problem, as long as the proof of 3-\Pi_1 does not mix the two symmetries up.