# The new dinv is not so new

## TL;DR

This paper reveals that the new dinv statistic introduced for the shuffle conjecture is equivalent to the traditional dinv in a specific case, providing new insights and proofs related to the Delta and shuffle conjectures.

## Contribution

It establishes a bijective equivalence between the ndinv and dinv statistics in a special case, and offers a non-compositional proof of the ehh case of the shuffle conjecture.

## Key findings

- ndinv matches dinv in a special case of the Delta conjecture
- Provides a bijective proof relating two-part and generalized Delta conjecture cases
- Offers a new non-compositional proof of the ehh case of the shuffle conjecture

## Abstract

In (Duane, Garsia, Zabrocki 2013) the authors introduced a new dinv statistic, denoted ndinv, on the two part case of the shuffle conjecture (Haglund et al. 2005) in order to prove a compositional refinement. Though in (Hicks, Kim 2013) a non-recursive (but algorithmic) definition of ndinv has been given, this statistic still looks a bit unnatural. In this paper we "unveil the mystery" around the ndinv, by showing bijectively that the ndinv actually matches the usual dinv statistic in a special case of the generalized Delta conjecture in (Haglund, Remmel, Wilson 2018). Moreover, we give also a non-compositional proof of the "$ehh$" case of the shuffle conjecture (after (Garsia, Xin, Zabrocki 2014)) by bijectively proving a relation with the two part case of the Delta conjecture.