Cyclic and lift closures for -avoiding permutations

We prove that the cyclic closure of the permutation class avoiding the pattern k ( k − 1 ) … 21 is finitely based. The minimal length of a minimal permutation is 2 k − 1 and these basis permutations are enumerated by ( 2 k − 1 ) . c k where c k is the k th Catalan number. We also define lift operations and give similar results. Finally, we consider the toric closure of a class and we propose some open problems.