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Toward Lower Bounds for Chromatic Symmetric Functions in the Elementary Basis

Isaiah Siegl

Abstract

Tatsuyuki Hikita recently proved the Stanley--Stembridge conjecture using probabilistic methods, showing that the chromatic symmetric functions of unit interval graphs are $e$-positive. Finding a combinatorial interpretation for these $e$-coefficients remains a major open problem. One approach is to look for combinatorial interpretations which are subsets of Gasharov's $P$-tableaux. Towards this goal, we introduce sets of strong and powerful $P$-tableaux, and use them to find combinatorial interpretations for various $e$-coefficients of the chromatic symmetric function $X_{inc(P)}(\mathbf{x}, q)$. We conjecture that the set of strong $P$-tableaux gives a lower bound for the $e$-coefficients of $X_{inc(P)}(\mathbf{x}, q)$. Additionally, we show that strong $P$-tableaux and the Shareshian--Wachs inversion statistic appear naturally in the proof of Hikita's result.

Toward Lower Bounds for Chromatic Symmetric Functions in the Elementary Basis

Abstract

Tatsuyuki Hikita recently proved the Stanley--Stembridge conjecture using probabilistic methods, showing that the chromatic symmetric functions of unit interval graphs are -positive. Finding a combinatorial interpretation for these -coefficients remains a major open problem. One approach is to look for combinatorial interpretations which are subsets of Gasharov's -tableaux. Towards this goal, we introduce sets of strong and powerful -tableaux, and use them to find combinatorial interpretations for various -coefficients of the chromatic symmetric function . We conjecture that the set of strong -tableaux gives a lower bound for the -coefficients of . Additionally, we show that strong -tableaux and the Shareshian--Wachs inversion statistic appear naturally in the proof of Hikita's result.

Paper Structure

This paper contains 15 sections, 50 theorems, 99 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Posets
  4. $P$-Tableaux
  5. Noncommutative $P$-Symmetric Functions
  6. Previous $m_\lambda^P(\mathbf{u})$ Positivity Results
  7. General $e$-Positivity Results
  8. Greedy Partition of $P$
  9. Positivity for the Path Graph
  10. Strong and Powerful $P$-Tableaux
  11. Tableaux Witnesses for Positivity
  12. Greedy Partition Coefficients and Tableaux Concatenation
  13. Positivity for $\lambda = (k-2,2)$
  14. Positivity of the Path Graph
  15. Examples and Data

Key Result

Theorem 1.1

HikitaChromatic Let $P$ be a $(\mathbf{3}+\mathbf{1})$ -free poset, and let $\text{\rm inc}(P)$ denote its incomparability graph. Then $X_{\text{\rm inc}(P)}(\mathbf{x}) = \sum_{\lambda \vdash |P|} c_\lambda^P e_{\lambda}(\mathbf{x})$ is $e$-positive. In other words, each elementary basis coefficien

Figures (9)

  • Figure 1: Ladder Swap
  • Figure 2:
  • Figure 3:
  • Figure 4:
  • Figure 5: Powerful $P_{\mathbf{m}}$-tableaux of shape $(3,2,2)$ with powersum words highlighted
  • ...and 4 more figures

Theorems & Definitions (124)

  • Theorem 1.1: Hikita's Theorem
  • Conjecture 1.2
  • Conjecture 1.3
  • Conjecture 1.4
  • Conjecture 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Theorem 1.9
  • Theorem 1.10
  • ...and 114 more