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 ${\mathrm{\Delta }}_{e_k}^{\prime }{e}_n$. Haglund et al. also propose the Extended Delta Conjecture for the delta operator expression ${\mathrm{\Delta }}_{e_k}^{\prime }{\mathrm{\Delta }}_{h_r}{e}_n$, 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 ${\mathrm{\Delta }}_{e_k}^{\prime }{\mathrm{\Delta }}_{h_r}{e}_n$ are equivalent when t or q equals 0, thus proving the valley version of the Extended Delta Conjecture when t or q equals 0.

Carlsson和Mellit的Shuffle定理给出了对角谐波环的Bigraded Frobenius特征的组合表达式，Haglund，Remmel和第二作者的Delta猜想为Delta算符表达式提供了Shuffle定理的两种推广 ${\mathrm{\Delta }}_{{Ë}_{ķ}}^{\prime }{Ë}_{ñ}$。Haglund等。还为增量算子表达式提出了扩展增量猜想${\mathrm{\Delta }}_{{Ë}_{ķ}}^{\prime }{\mathrm{\Delta }}_{H_{\mathrm{[R}}}{Ë}_{ñ}$，类似于Delta Conjecture的上升版本。最近，D'Adderio，Iraci和Wyngaerd在以下情况下证明了扩展Delta猜想的上升版本：$Ť=0$。在本文中，我们提出了扩展三角洲猜想的新谷型。然后，我们研究扩展有序多集分区的组合，以证明两个猜想${\mathrm{\Delta }}_{{Ë}_{ķ}}^{\prime }{\mathrm{\Delta }}_{H_{\mathrm{[R}}}{Ë}_{ñ}$当t或q等于0时是等价的，因此证明了当t或q等于0时扩展Delta猜想的谷型。