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Math-2419 · mathematics · Graph theory

Stanley–Stembridge Chromatic Symmetric Function Conjecture

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Statement

Definitions and mathematical statement

Shared notation.
$\mathbb N$, $\mathbb Z$, $\mathbb Q$, $\mathbb R$, and $\mathbb C$ denote the natural numbers, integers, rationals, reals, and complex numbers, respectively. All additional parameters and hypotheses are specified in the question.
Statement recorded in the supplied source.
Stanley–Stembridge conjecture: if $G=G(m)$ is the incomparability graph associated with a Hessenberg function $m$, then its chromatic symmetric function $X_G(x)$ is a nonnegative sum of elementary symmetric functions.

Formulation references: [S1], [S2].

Short English statement

Stanley–Stembridge conjecture: if $G=G(m)$ is the incomparability graph associated with a Hessenberg function $m$, then its chromatic symmetric function $X_G(x)$ is a nonnegative sum of elementary symmetric functions.

Sources

Catalogue entry

Statement licence: CC-BY-SA-4.0 · Source file

Contributions 1

Math-2419-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2419

Elementary positivity for unit interval graphs: The theorem proves coefficientwise elementary positivity in q for all natural unit interval graphs G(m). At q=1 this is the complete recorded chromatic-symmetric-function assertion; the stated Guay-Paquet reduction also covers all (3+1)-free posets.

The claimed result implies the stated question and also supplies the additional strength described above.

Stanley–Stembridge Chromatic Symmetric Function Conjecture

OpenAI's claim

Elementary positivity for unit interval graphs: OpenAI claims nonnegative $\mathbb N[q]$ coefficients in the elementary-basis expansion of chromatic quasisymmetric functions of natural unit interval graphs. This concerns the elementary-positivity portion of the Shareshian--Wachs conjecture.

Mathematical statement of the claimed result

Mathematical claim 1.
For weakly increasing integers $i\le h(i)\le n$, define $G$ on $[n]$ by $\{i,j\}\in E(G)$ exactly when $i<j\le h(i)$. Its chromatic quasisymmetric function is $\chi_G(X;q)=\sum_{f\text{ proper}}q^{\#\{ij\in E:i<j,\ f(i)<f(j)\}}\prod_i x_{f(i)}$. The claim is a formal identity \[\chi_G(X;q)=\sum_{\sigma\in D_G^0}q^{\operatorname{inv}_G(\sigma)}e_{\theta_G(\sigma)}(X),\] where $D_G^0$ consists of permutations whose adjacent descents are graph edges, $\operatorname{inv}_G$ counts inversions whose values form edges, and a deterministic terminating procedure supplies a partition $\theta_G(\sigma)\vdash n$. Since $e_\lambda=\prod_i e_{\lambda_i}$, all elementary-basis coefficients are polynomials in $q$ with nonnegative integer coefficients.

Outline of the argument

Elementary positivity for unit interval graphs: Positive wall transformations produce sections indexed by exponent-lattice elements. Interval rotations identify the relevant graph functions with single positive sections, and a support argument forces the contributing exponents onto distinguished anchors. A packet bijection then extracts a nonnegative combinatorial expansion whose powers of $q$ record the accumulated jumps.

Formal verification

Family 169 describes Lean coverage for a nonnegative elementary expansion for natural unit interval graphs with its inversion weights.

Elementary positivity for unit interval graphs: The manuscript links ElementaryPositivity.lean.

The documented selection includes this manuscript’s main assertion; that assertion retains the restrictions stated above.

This site has not run the Lean development. These links describe OpenAI’s selected formal statements and should be read with their stated scope.

Status caveat

OpenAI reports that an internal model produced the manuscripts. Their arguments have not been independently verified as proofs by this site. Any portion outside the documented formal scope may contain mistakes, and the catalogue label reports the announced mathematical claim.

Sources

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