P4552 differential

The first question is easy to see $max(p,n)$

Second questions , consider $b[1]$ How many possible results can be formed by us .

The essence of making every number the same is to make the last of the difference group $n-1$ All items become 0.
Every step of the process Namely Let a positive number –, negative ++.

Because it is necessary to ensure the minimum number of steps , Then there is at least $min(p,n)$ The first step is from $[2,n]$ Take out a pair at the same time +1-1

be left over $abs(p-n)$ Step , There are two kinds of decisions , Either with $b[1]$ operation , Either with $b[n+1]$ operation .
And because there is no difference in operation , We only look at the final result .$b[1]$ The value range of is $[b[1],b[1]+abs(p-n)]$.

Advanced :

Ask how many different solutions make them the same .

Any two schemes are different , The operation of a certain step is different