Generalized pattern avoidance with additional restrictions

TL;DR

This paper studies permutations avoiding certain generalized patterns with additional restrictions, providing combinatorial bijections, generating functions, and connections to set partitions and trees, thus extending pattern avoidance theory.

Contribution

It introduces new combinatorial bijections, derives generating functions, and connects pattern-avoiding permutations to set partitions and tree structures, advancing the understanding of generalized pattern avoidance.

Findings

Number of permutations avoiding 1-32 with increasing last k letters relates to Bell numbers.

Derived exponential generating functions for permutations avoiding specific generalized patterns.

Established bijections between pattern-avoiding permutations and set partitions or trees.

Abstract

Babson and Steingr\'{\i}msson introduced generalized permutation patterns that allow the requirement that two adjacent letters in a pattern must be adjacent in the permutation. We consider n-permutations that avoid the generalized pattern 1-32 and whose k rightmost letters form an increasing subword. The number of such permutations is a linear combination of Bell numbers. We find a bijection between these permutations and all partitions of an $(n-1)$-element set with one subset marked that satisfy certain additional conditions. Also we find the e.g.f. for the number of permutations that avoid a generalized 3-pattern with no dashes and whose k leftmost or k rightmost letters form either an increasing or decreasing subword. Moreover, we find a bijection between n-permutations that avoid the pattern 132 and begin with the pattern 12 and increasing rooted trimmed trees with n+1 nodes.