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Smirnov words and the Delta Conjectures

Alessandro Iraci, Philippe Nadeau, Anna Vanden Wyngaerd

Abstract

We provide a combinatorial interpretation of the symmetric function $\left.Θ_{e_k}Θ_{e_l}\nabla e_{n-k-l}\right|_{t=0}$ in terms of segmented Smirnov words. The motivation for this work is the study of a diagonal coinvariant ring with one set of commuting and two sets of anti-commuting variables, whose Frobenius characteristic is conjectured to be the symmetric function in question. Furthermore, this function is related to the Delta conjectures. Our work is a step towards a unified formulation of the two versions, as we prove a unified Delta theorem at $t=0$.

Smirnov words and the Delta Conjectures

Abstract

We provide a combinatorial interpretation of the symmetric function in terms of segmented Smirnov words. The motivation for this work is the study of a diagonal coinvariant ring with one set of commuting and two sets of anti-commuting variables, whose Frobenius characteristic is conjectured to be the symmetric function in question. Furthermore, this function is related to the Delta conjectures. Our work is a step towards a unified formulation of the two versions, as we prove a unified Delta theorem at .
Paper Structure (18 sections, 28 theorems, 77 equations, 4 figures)

This paper contains 18 sections, 28 theorems, 77 equations, 4 figures.

Table of Contents

  1. Combinatorial definitions
  2. Algebraic recursion
  3. Combinatorial expansion of the main symmetric function
  4. The standard case
  5. The general case
  6. Unified Delta theorem at t=0
  7. The Delta conjecture
  8. From segmented Smirnov words to decorated Dyck paths
  9. Another statistic on segmented Smirnov words
  10. Equidistribution of sminv and sdinv
  11. Unified Delta theorem at t=0
  12. 'Catalan' case via fundamental quasisymmetric expansion
  13. Fundamental quasisymmetric expansion
  14. The 'Catalan' case
  15. The maximal case k+l=n-1
  16. ...and 3 more sections

Key Result

Proposition 1.11

The map $\Pi$ restricts to bijections sending $\mathsf{sminv}$ to $\mathsf{inv}$ in both cases.

Figures (4)

  • Figure 1: An element of $\mathsf{LD}(8)^{\ast 2, \bullet 2}$ (left) and an element of $\mathsf{LD}_0(8)^{\ast 4,\bullet 2}$ (right)
  • Figure 2: The images of the ordered set partition $34|1|24|124$ (or $43|1|42|421$) via the two bijections.
  • Figure 3: Four cases for the occurrence of the maximal label $m$, and how to delete it.
  • Figure 4: Two labelled parallelogram polyominoes of size $6 \times 4$, the right one has area $0$.

Theorems & Definitions (83)

  • Definition 1.1
  • Example 1.2
  • Definition 1.3
  • Definition 1.4
  • Definition 1.5
  • Definition 1.6
  • Definition 1.7
  • Remark 1.8
  • Example 1.9
  • Definition 1.10
  • ...and 73 more