Self-modified difference ascent sequences

TL;DR

This paper introduces and characterizes self-modified difference ascent sequences, a generalization of ascent sequences, and enumerates them using generalized Fibonacci polynomials, linking them to $d$-Fishburn permutations.

Contribution

It extends the hat map to difference ascent sequences, characterizes their fixed points, and provides enumeration formulas involving generalized Fibonacci polynomials.

Findings

Characterization of self-modified difference ascent sequences

Enumeration using generalized Fibonacci polynomials

Description of related $d$-Fishburn permutations

Abstract

Ascent sequences play a key role in the combinatorics of Fishburn structures. Difference ascent sequences are a natural generalization obtained by replacing ascents with $d$-ascents. We have recently extended the so-called hat map to difference ascent sequences, and self-modified difference ascent sequences are the fixed points under this map. We characterize self-modified difference ascent sequences and enumerate them in terms of certain generalized Fibonacci polynomials. Furthermore, we describe the corresponding subset of $d$-Fishburn permutations.