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The valley version of the Extended Delta Conjecture

Dun Qiu, Andrew Timothy Wilson

TL;DR

This work introduces the valley version of the Extended Delta Conjecture for $Δ'_{e_k}Δ_{h_r}e_n$ and develops extended ordered multiset partitions as the central combinatorial model. It establishes equidistribution of the statistics $\mathrm{inv}$, $\mathrm{maj}$, $\mathrm{dinv}$, and $\mathrm{minimaj}$ via explicit insertion maps, enabling a valley-case proof when $t=0$ or $q=0$ and linking to the rise version. The paper derives recursive descriptions for the inv and minimaj generating functions, connecting to Mahonian distributions and $q$-Stirling numbers, and discusses Schur-positivity through LLT and crystal-theoretic perspectives. It also outlines the broader implications for the Extended Delta Conjecture, including equivalences at specializations, and sketches future conjectures, notably involving hook-shaped $\lambda$ in delta operators. Overall, the results advance the combinatorial understanding of valley structures in Delta-type conjectures and pave the way for full general proofs and positivity theorems.

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 $Δ'_{e_k} e_n$. Haglund et al. also propose the Extended Delta Conjecture for the delta operator expression $Δ'_{e_k} Δ_{h_r}e_n$, 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 $Δ'_{e_k} Δ_{h_r}e_n$ are equivalent when $t$ or $q$ equals 0, thus proving the valley version of the Extended Delta Conjecture when $t$ or $q$ equals 0.

The valley version of the Extended Delta Conjecture

TL;DR

This work introduces the valley version of the Extended Delta Conjecture for and develops extended ordered multiset partitions as the central combinatorial model. It establishes equidistribution of the statistics , , , and via explicit insertion maps, enabling a valley-case proof when or and linking to the rise version. The paper derives recursive descriptions for the inv and minimaj generating functions, connecting to Mahonian distributions and -Stirling numbers, and discusses Schur-positivity through LLT and crystal-theoretic perspectives. It also outlines the broader implications for the Extended Delta Conjecture, including equivalences at specializations, and sketches future conjectures, notably involving hook-shaped in delta operators. Overall, the results advance the combinatorial understanding of valley structures in Delta-type conjectures and pave the way for full general proofs and positivity theorems.

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 . Haglund et al. also propose the Extended Delta Conjecture for the delta operator expression , 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 . 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 are equivalent when or equals 0, thus proving the valley version of the Extended Delta Conjecture when or equals 0.

Paper Structure

This paper contains 23 sections, 21 theorems, 88 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Extended ordered multiset partitions
  4. Ordered set partitions and ordered multiset partitions
  5. Extended permutations, extended ordered set and multiset partitions
  6. The identity $D_{r;\beta,k}^{\mathrm{dinv}}(q) =D_{r;\beta,k}^{\mathrm{maj}}(q) =D_{r;\beta,k}^{\mathrm{inv}}(q)$
  7. The insertion map for inv
  8. The insertion map for maj
  9. The insertion map for dinv
  10. The identity $D_{r;\beta,k}^{\mathrm{inv}}(q) =D_{r;\beta,k}^{\mathrm{minimaj}}(q)$
  11. The recursion for inv
  12. The recursion for minimaj
  13. Conclusion and future directions
  14. The shared distribution
  15. Schur positivity
  16. ...and 8 more sections

Key Result

Theorem 1.1

For any integer $n\geq 0$,

Figures (10)

  • Figure 1: The Ferrers diagram of the partition $\lambda=(7,7,5,3,3)$.
  • Figure 2: A partition $\mu=(3,1)$.
  • Figure 3: A $(7,7)$-parking function with area 13 and dinv 2.
  • Figure 4: Examples: parking functions in $\mathcal{WPF}_{7,2}^{\mathrm{Rise}}$ and $\mathcal{WPF}_{7,2}^{\mathrm{Val}}$.
  • Figure 5: A $(7,7)$-extended parking function with $2$ blank valleys.
  • ...and 5 more figures

Theorems & Definitions (28)

  • Theorem 1.1: Carlsson and Mellit
  • Conjecture 1.1: Haglund, Remmel and Wilson
  • Conjecture 1.2: Rise version of the Extended Delta Conjecture HRW
  • Conjecture 1.3: Valley version of the Extended Delta Conjecture
  • Theorem 3.1: Haglund, Remmel and Wilson
  • Theorem 3.2: Rhoades and Wilson
  • Theorem 3.3
  • proof
  • Corollary 4.1
  • Theorem 4.2
  • ...and 18 more