# The valley version of the Extended Delta Conjecture

## TL;DR

This paper introduces a new valley version of the Extended Delta Conjecture, proving its validity at specific parameter values and establishing its equivalence to the rise version in those cases.

## Contribution

It proposes a novel valley version of the Extended Delta Conjecture and proves its validity at t=0 or q=0, linking it to existing rise version results.

## Key findings

- Proposed a valley version of the Extended Delta Conjecture.
- Proved the conjecture when t=0 or q=0.
- Established equivalence with the rise version at these parameter values.

## Abstract

The Shuffle Theorem of Carlsson and Mellit gives a combinatorial expression for the bigraded Frobenius characteristic of the ring of diagonal harmonics, and the Delta Conjecture of Haglund, Remmel and the second author provides two generalizations of the Shuffle Theorem to the delta operator expression $\Delta'_{e_k} e_n$. Haglund et al. also propose the Extended Delta Conjecture for the delta operator expression $\Delta'_{e_k} \Delta_{h_r}e_n$, which is analogous to the rise version of the Delta Conjecture. Recently, D'Adderio, Iraci and Wyngaerd proved the rise version of the Extended Delta Conjecture at the case when $t=0$. In this paper, we propose a new valley version of the Extended Delta Conjecture. Then, we work on the combinatorics of extended ordered multiset partitions to prove that the two conjectures for $\Delta'_{e_k} \Delta_{h_r}e_n$ are equivalent when $t$ or $q$ equals 0, thus proving the valley version of the Extended Delta Conjecture when $t$ or $q$ equals 0.