Weak ascent sequences and related combinatorial structures

TL;DR

This paper introduces weak ascent sequences, a new class of sequences that encode various combinatorial objects and are related to pattern-avoiding sequences, providing enumeration formulas and structural insights.

Contribution

The paper defines weak ascent sequences, shows their encoding of multiple combinatorial structures, and derives a closed-form enumeration formula for them.

Findings

Weak ascent sequences encode pattern-avoiding permutations and matrices.

A closed-form enumeration formula for weak ascent sequences is provided.

Weak ascent sequences relate to recent research on inversion sequences.

Abstract

In this paper we introduce {\em weak ascent sequences}, a class of number sequences that properly contains ascent sequences. We show how these sequences uniquely encode each of the following objects: permutations avoiding a particular length-4 bivincular pattern; upper-triangular binary matrices that satisfy a column-adjacency rule; factorial posets that are weakly (3+1)-free. We also show how weak ascent sequences are related to a class of pattern avoiding inversion sequences that has been a topic of recent research by Auli and Elizalde. Finally, we consider the problem of enumerating these new sequences and give a closed form expression for the number of weak ascent sequences having a prescribed length and number of weak ascents.