Fishburn trees

TL;DR

This paper introduces Fishburn trees, a special class of labeled binary trees linked to Fishburn numbers, establishing bijections with known combinatorial structures and applying these results to problems on ascent sequences.

Contribution

It defines Fishburn trees, explores their bijections with other structures, and applies these to solve problems related to ascent sequences.

Findings

Fishburn trees are bijective with several combinatorial structures.

New simplified maps between Fishburn trees and other structures.

Applications to flip and sum problems on ascent sequences.

Abstract

The in-order traversal provides a natural correspondence between binary trees with a decreasing vertex labeling and endofunctions on a finite set. By suitably restricting the vertex labeling we arrive at a class of trees that we call Fishburn trees. We give bijections between Fishburn trees and other well-known combinatorial structures that are counted by the Fishburn numbers, and by composing these new maps we obtain simplified versions of some of the known maps. Finally, we apply this new machinery to the so called flip and sum problems on modified ascent sequences.