Avoiding patterns in irreducible permutations

Jean-Luc Baril (1)

1 Le2i - Laboratoire Electronique, Informatique et Image [UMR6306]

We explore the classical pattern avoidance question in the case of irreducible permutations, i.e., 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.

Mots clés: Pattern avoiding permutation; irreducible permutation; succession; involution; Motzkin path

Domaines: Mathématique discrète [cs.DM]

Jean-Luc Baril. Avoiding patterns in irreducible permutations. Discrete Mathematics and Theoretical Computer Science, 2016, Vol. 17 no. 3 (3), pp.13-30. ⟨10.46298/dmtcs.2158⟩. ⟨hal-01352852⟩