Pattern avoidance in flattened derangements

To flatten a permutation , expressed in standard cycle form, is to remove the parentheses enclosing the cycles and consider the resulting permutation in the one-line notation.Then is said to avoid a pattern in the flattened sense if avoids in the usual sense.In this paper, we consider the problem of avoidance of one or more classical patterns of length three in the flattened sense by derangements, which extends earlier results on flattened permutations and other structures.We establish explicit formulas enumerating each corresponding avoidance class of derangements according to the number of cycles.As a consequence of our results, we obtain the equivalences 213 312 and 231 321 for derangements in the flattened sense.To establish the generating function formula in the case of the pattern 321, we make use of the kernel method and Lagrange inversion.