
In This Way, but Nicely. Pattern Avoidance. Follow-Up. 227
11. Prove that S
n
(1342, 2431) is a P -recursive sequence.
12. Prove that the sequence f(n)=(n!)
n
is not P -recursive.
13. (–) Characterize layered permutations by pattern avoidance.
14. Find the packing density of the pattern q
k
=1(k +1)k ···2.
15. Prove that for any nonnegative integer s with s ≤
n
2
there is a per-
mutation p ∈ S
n
having s copies of the pattern 21 and no copies of
132.
16. Let N be a positive integer. Show that there exists a pattern q and a
positive integer n so that the frequency sequence (S
n,q
(c))
c≥0
contains
N consecutive internal zeros.
17. (–) Let p be a 321-avoiding n-per