Simple permutations of the classes Av(321, 3412) and Av(321, 4123) have polynomial growth

A permutation is called simple if its only blocks i.e. subsets of the permutation consist of singleton and the permutation itself. For example, 2134 is not a simple permutation since it consists of a block 213 but 3142 is a simple permutation. The basis of a class of permutations is a set of patterns, which is minimal under involvement and do not belong to the permutation. In this paper we prove that the number of simple permutations an of the pattern class Av(321, 3412) follows the recurrence an = an−1+an−2 for n ≥ 4 and the pattern class Av(321, 4123) follows the recurrence an = an−2+an−3 for n ≥ 7. Thus, these pattern classes have polynomial growth.