The valley version of the Extended Delta Conjecture

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.