Ordered set partition statistics and the Delta Conjecture

TL;DR

This paper provides evidence for the Delta Conjecture by proving equidistribution results for statistics on ordered set partitions, extending classical permutation results to new combinatorial objects.

Contribution

It proves conjectures related to ordered set partition statistics, supporting the Delta Conjecture and generalizing MacMahon's classical permutation result.

Findings

Proves equidistribution of inversion count and major index on ordered set partitions.

Supports the Delta Conjecture with new combinatorial evidence.

Extends classical permutation statistics to ordered set partitions.

Abstract

The Delta Conjecture of Haglund, Remmel, and Wilson is a recent generalization of the Shuffle Conjecture in the field of diagonal harmonics. In this paper we give evidence for the Delta Conjecture by proving a pair of conjectures of Wilson and Haglund-Remmel-Wilson which give equidistribution results for statistics related to inversion count and major index on objects related to ordered set partitions. Our results generalize the famous result of MacMahon that major index and inversion number share the same distribution on permutations.