Hyperplane arrangement : The Shi arrangement

Question

I have been lately reading Hyperplane arrangement lectures by Richard Stanley on https://www.cis.upenn.edu/~cis610/sp06stanley.pdf . In lecture 5, Theorem 5.16 we define the characteristic polynomial of the Shi arrangement by $$X_{S_n}(t)=t(t-n)^{n-1}$$ I am struggling to understand part of the proof of the theorem. More precisely, why did we partition [n] into p-n blocks ? How did we know that the maximum number of blocks that we can have is p-n ? I think the rest of the proof is clear. I am just trying to know why we thought of p-n. Any ideas ?