Bijections for inversion sequences, ascent sequences and 3-nonnesting set partitions

TL;DR

This paper establishes a bijective proof connecting inversion sequences and 3-nonnesting set partitions, confirming a conjecture and providing new combinatorial insights into these structures.

Contribution

It offers a novel bijective proof linking inversion sequences and 3-nonnesting partitions, advancing combinatorial understanding of these objects.

Findings

Confirmed the conjecture on the cardinality equivalence.

Provided a new bijective proof for a conjecture by Duncan and Steingrímsson.

Connected different combinatorial structures through explicit bijections.

Abstract

Set partitions avoiding $k$-crossing and $k$-nesting have been extensively studied from the aspects of both combinatorics and mathematical biology. By using the generating tree technique, the obstinate kernel method and Zeilberger's algorithm, Lin confirmed a conjecture due independently to the author and Martinez-Savage that asserts inversion sequences with no weakly decreasing subsequence of length 3 and enhanced 3-nonnesting partitions have the same cardinality. In this paper, we provide a bijective proof of this conjecture. Our bijection also enables us to provide a new bijective proof of a conjecture posed by Duncan and Steingr\'{\i}msson, which was proved by the author via an intermediate structure of growth diagrams for $01$-fillings of Ferrers shapes.