Avoiding patterns in irreducible permutations

International audienceWe 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

Distributions of several infinite families of mesh patterns

Br\"and\'en and Claesson introduced mesh patterns to provide explicit expansions for certain permutation statistics as linear combinations of (classical) permutation patterns. The first systematic study of avoidance of mesh patterns was conducted by Hilmarsson et al., while the first systematic study of the distribution of mesh patterns was conducted by the first two authors. In this paper, we provide far-reaching generalizations for 8 known distribution results and 5 known avoidance results related to mesh patterns by giving distribution or avoidance formulas for certain infinite families of mesh patterns in terms of distribution or avoidance formulas for smaller patterns. Moreover, as a corollary to a general result, we find the distribution of one more mesh pattern of length 2.Comment: 27 page

Pattern-avoiding alternating words

A word $w=w_1w_2\cdots w_n$ is alternating if either $w_1w_3\cdots$ (when the word is up-down) or $w_1>w_2w_4<\cdots$ (when the word is down-up). In this paper, we initiate the study of (pattern-avoiding) alternating words. We enumerate up-down (equivalently, down-up) words via finding a bijection with order ideals of a certain poset. Further, we show that the number of 123-avoiding up-down words of even length is given by the Narayana numbers, which is also the case, shown by us bijectively, with 132-avoiding up-down words of even length. We also give formulas for enumerating all other cases of avoidance of a permutation pattern of length 3 on alternating words