Modified difference ascent sequences and Fishburn structures

TL;DR

This paper introduces modified difference ascent sequences and extends the hat map to inversion sequences, revealing new combinatorial structures related to Fishburn numbers and permutation classes.

Contribution

It defines modified difference ascent sequences, extends the hat map to inversion sequences, and identifies new permutation sets counted by Fishburn numbers.

Findings

Introduction of modified difference ascent sequences

Extension of the hat map to inversion sequences

Identification of new Fishburn-numbered permutation classes

Abstract

Ascent sequences and their modified version play a central role in the bijective framework relating several combinatorial structures counted by the Fishburn numbers. Ascent sequences are positive integer sequences defined by imposing a bound on the growth of their entries in terms of the number of ascents contained in the corresponding prefix, while modified ascent sequences are the image of ascent sequences under the so-called hat map. By relaxing the notion of ascent, Dukes and Sagan have recently introduced difference ascent sequences. Here we define modified difference ascent sequences and study their combinatorial properties. Inversion sequences are a superset of the difference ascent sequences and we extend the hat map to this domain. Our extension depends on a parameter which we specialize to obtain a new set of permutations counted by the Fishburn numbers and characterized by a subdiagonality property.