Abstract
We explore the classical pattern avoidance question in the case of irreducible permutations, <i>i.e.</i>, those in which there is no index $i$ such that $\sigma (i+1) - \sigma (i)=1$. The problem is addressed completely in the case of avoiding one or two patterns of length three, and several well known sequences are encountered in the process, such as Catalan, Motzkin, Fibonacci, Tribonacci, Padovan and Binary numbers. Also, we present constructive bijections between the set of Motzkin paths of length $n-1$ and the sets of irreducible permutations of length $n$ (respectively fixed point free irreducible involutions of length $2n$) avoiding a pattern $\alpha$ for $\alpha \in \{132,213,321\}$. This induces two new bijections between the set of Dyck paths and some restricted sets of permutations.
Highlights
Introduction and notationLet Sn be the set of permutations on [n] = {1, 2, . . . , n}, i.e., all one-to-one correspondences from [n] into itself
We present constructive bijections between the set of Motzkin paths of length n − 1 and the sets of irreducible permutations of length n avoiding a pattern α for α ∈ {132, 213, 321}
Many classical sequences in combinatorics appear as the cardinality of pattern-avoiding permutation classes
Summary
The problem is addressed completely in the case of avoiding one or two patterns of length three, and several well-known sequences are encountered, such as the Catalan, Motzkin, Fibonacci, Binary, Tribonacci and Padovan numbers. Generalizing the two concepts of inflation and deflation for sets of fixed point free involutions, we obtain similar results for involutions. Exhaustive enumerative results are obtained, and we construct bijections between fixed point free irreducible involutions of length 2n avoiding α ∈ {132, 213, 321} and Motzkin paths of length n − 1
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