Cyclic and lift closures for k…21k…21-avoiding permutations

We prove that the cyclic closure of the permutation class avoiding the pattern k(k−1)…21k(k−1)…21 is finitely based. The minimal length of a minimal permutation is 2k−12k−1 and these basis permutations are enumerated by (2k−1).ck(2k−1).ck where ckck is the kkth 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.