Pattern avoidance in flattened permutations Toufik Mansour Department of Mathematics, University of Haifa, 31905 Haifa, Israel toufik@math.haifa.ac.il Mark Shattuck Department of Mathematics, University of Haifa, 31905 Haifa, Israel maarkons@excite.com, shattuck@math.utk.edu Abstract To flatten a permutation expressed as a product of disjoint cycles, we mean to form another permutation by erasing the parentheses which enclose the cycles of the original. This clearly depends on how the cycles are listed. For permutations written in the standard cycle form–cycles arranged in increasing order of their first entries, with the smallest element first in each cycle–we count the permutations of [n] whose flattening avoids any subset of S3. Among the sequences that arise are central binomial coefficients, Schr¨oder numbers, and relatives of the Fibonacci numbers. In some instances, we provide combinatorial arguments of the result, while in others, our approach is more algebraic. In a couple of the cases, we define an explicit bijection between the subset of Sn in question and a restricted set of lattice paths. In another, to establish the result, we make use of the kernel method to solve a functional equation arising once a certain parameter has been considered. Keywords: pattern avoidance, permutation. 2010 Mathematics Subject Classification: 05A15, 05A05. 1 Introduction We will use the following notational conventions: N = {0, 1, 2,... }, P = {1, 2,...},[n]= {1, 2,...,n} for n ∈ P, and [m, n]= {m, m +1,...,n} for m, n ∈ P, with m ≤ n. By con- vention, we take [0] = ∅ and [m, n]= ∅ if m>n. A permutation σ of [n] is often represented as a word σ = σ 1 σ 2 ··· σ n wherein σ(i)= σ i for each i or as a product of disjoint cycles (a 1 ··· a r )(b 1 ··· b s ) ··· wherein the first cycle, for example, we have σ(a i )= a i+1 ,1 ≤ i ≤ r - 1, and σ(a r )= a 1 . A permutation expressed as a product of disjoint cycles is said to be in stan- dard cycle form if the cycles are written with the smallest element as the first entry, with cycles arranged in ascending order according to first entries. The pattern avoidance problem for permutations has been studied extensively from various perspectives in both enumerative and algebraic combinatorics, see, e.g., [4]. The comparable problem has also been considered on other structures such as words, compositions, and set partitions. Here, we consider a notion of pattern avoidance for permutations which is analogous to the one recently introduced by Callan [2] for set partitions. Suppose σ is a permutation of [n] written in standard cycle form. Define Flatten(σ) to be the permutation of [n] obtained by erasing the parentheses enclosing the cycles of σ. For example, if σ = 71564328 ∈ S 8 , then the standard form is (172)(3546)(8) and Flatten(σ) = 17235468. Here, we will say that a permutation σ contains an occurrence of the permutation pattern ρ if Flatten(σ) contains a subsequence isomorphic to ρ (i.e., a subsequence which, when standardized, gives ρ). Otherwise, we will say that σ avoids ρ. For example, the permutation 56142387 = 1