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$e$-Log-Concavity of Chromatic Quasisymmetric Functions

Chromatic quasisymmetric functions of natural unit interval graphs were conjectured to have log-concave coefficients in the elementary basis. A connected $13$-vertex example refutes it: for the Hessenberg function $h=(2,4,4,6,7,10,10,10,10,12,12,13,13)$ and $\lambda=(6,5,1,1)$ the coefficients of $q^5,q^6,q^7$ are $1,6,38$, and $6^2 < 1 \cdot 38$. The coefficient is still positive, palindromic and unimodal, so log-concavity is separated from the weaker shape properties that motivated the conjecture.

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combinatoricsJul 22, 2026Significance 15/100Registry: site confirmed

$e$-Log-Concavity of Chromatic Quasisymmetric Functions

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Chromatic quasisymmetric functions of natural unit interval graphs were conjectured to have log-concave coefficients in the elementary basis. A connected $13$-vertex example refutes it: for the Hessenberg function $h=(2,4,4,6,7,10,10,10,10,12,12,13,13)$ and $\lambda=(6,5,1,1)$ the coefficients of $q^5,q^6,q^7$ are $1,6,38$, and $6^2 < 1 \cdot 38$. The coefficient is still positive, palindromic and unimodal, so log…

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Chromatic quasisymmetric functions of natural unit interval graphs were conjectured to have log-concave coefficients in the elementary basis. A connected $13$-vertex example refutes it: for the Hessenberg function $h=(2,4,4,6,7,10,10,10,10,12,12,13,13)$ and $\lambda=(6,5,1,1)$ the coefficients of $q^5,q^6,q^7$ are $1,6,38$, and $6^2 < 1 \cdot 38$. The coefficient is still positive, palindromic and unimodal, so log-concavity is separated from the weaker shape properties that motivated the conjecture.

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