Simple permutations of the classes Av(321, 13524) and Av(321, 13452) 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 permutation is a pattern 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 with two basis of length 3 and 5 such as Av(321, 13452) and Av(321, 13524) have polynomial growth.