Transport of patterns by Burge transpose

TL;DR

This paper introduces the Burge transpose, a novel operation for transporting pattern avoidance properties between Fishburn permutations, ascent sequences, and Cayley permutations, revealing new equivalences and connections.

Contribution

It develops a general framework for pattern transport using the Burge transpose, linking various permutation classes and introducing a new perspective on pattern equivalences.

Findings

Burge transpose effectively transports pattern avoidance between permutation classes.

Established a Wilf-equivalence framework for Cayley permutations using mesh patterns.

Connected primitive ascent sequences with pattern transport mechanisms.

Abstract

We take the first steps in developing a theory of transport of patterns from Fishburn permutations to (modified) ascent sequences. Given a set of pattern avoiding Fishburn permutations, we provide an explicit construction for the basis of the corresponding set of modified ascent sequences. Our approach is in fact more general and can transport patterns between permutations and equivalence classes of so called Cayley permutations. This transport of patterns relies on a simple operation we call the Burge transpose. It operates on certain biwords called Burge words. Moreover, using mesh patterns on Cayley permutations, we present an alternative view of the transport of patterns as a Wilf-equivalence between subsets of Cayley permutations. We also highlight a connection with primitive ascent sequences.