Each will have some number of local maxima, m.

For example if n=6 some permutations are
m=1 123465
m=2 143256
m=3 214356

Define f(n,m) as the number of permutations of (1,2,3...n) with m local maxima.

What may be ultimately sought is a formula for f(n,m) but here are some simpler considerations to prove:

Find a formula for f(n,2) in terms of values where m=1

Find a formula relating f(2a,a) and f(2a+1,2a).

Note: The problem of finding f(n,1) was investigated here