1 IntroductionLet q = q1


qm be a finite string of numbers. The reduction of q, denoted red(q), isthe string obtained by replacing the ith smallest letter(s) of q with i. For example, thered(2674425) = 1452213. Given two permutations p ∈ Sn, q ∈ Sm, we say p contains qas a pattern if there exist 1 6 i1 1


pim ) = q. Otherwisep avoids q. This definition of pattern avoidance appears in the characterization of 1-stack sortable permutations and the characterization of smooth Schubert varieties. Further, it introduces an interesting and well-studied enumeration problem; namely,count the elements of the set Sn(Q) = { p ∈ Sn | p avoids q for all q ∈ Q} .The focus of this paper is not the study Sn(Q), but a variation of pattern avoidancegiven by the following definitions. Given q, the barred permutation qis the permutation obtained by copying the entries of qand putting a bar over...