Ascent sequences and upper triangular matrices containing non-negative integers

TL;DR

This paper establishes a bijection between ascent sequences and a class of upper triangular matrices with non-negative entries, revealing new combinatorial equivalences and characterizations of special matrix classes.

Contribution

It introduces a novel bijection linking ascent sequences with upper triangular matrices and explores their combinatorial properties and statistics.

Findings

Equivalence of natural statistics under the bijection

Binary matrices correspond to ascent sequences with no consecutive equal entries

Bidiagonal matrices relate to order-consecutive set partitions

Abstract

This paper presents a bijection between ascent sequences and upper triangular matrices whose non-negative entries are such that all rows and columns contain at least one non-zero entry. We show the equivalence of several natural statistics on these structures under this bijection and prove that some of these statistics are equidistributed. Several special classes of matrices are shown to have simple formulations in terms of ascent sequences. Binary matrices are shown to correspond to ascent sequences with no two adjacent entries the same. Bidiagonal matrices are shown to be related to order-consecutive set partitions and a simple condition on the ascent sequences generate this class.