Permutations Restricted By Primes

[Leroy Quet]

I was wondering, how many permutations <n(1),n(2),...,n(m)> are there of
<1,2,...,m>, where

n(k) < n(k-1) whenever k is prime,
n(k) > n(k-1) whenever k is composite?

For example, for m = 4 we have the permutations:
3, 2, 1, 4
4, 3, 1, 2
4, 2, 1, 3


So the 4th term of the sequence giving the number of permutations would 
be 3.


I do not know if this sequence is in the EIS.


Another possible sequence would be the number of permutations, where (as 
above)
n(k) < n(k-1) whenever k is prime,
but there is no restriction on n(k) if k is composite.

And of course, we can use other criteria besides primeness to generate 
other permutation-counting sequences using the same basic idea.

thanks,
Leroy Quet