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Abreu, Alex; Nigro, Antonio

Chromatic symmetric functions from the modular law. (English) Zbl 1459.05334

J. Comb. Theory, Ser. A 180, Article ID 105407, 31 p. (2021).
Summary: In this article we show how to compute the chromatic quasisymmetric function of indifference graphs from the modular law introduced in [M. Guay-Paquet, “A modular relation for the chromatic symmetric functions of (3+1)-free posets”, Preprint, arXiv:1306.2400]. We provide an algorithm which works for any function that satisfies this law, such as unicellular LLT polynomials. When the indifference graph has bipartite complement it reduces to a planar network, in this case, we prove that the coefficients of the chromatic quasisymmetric function in the elementary basis are positive unimodal polynomials and characterize them as certain \(q\)-hit numbers (up to a factor). Finally, we discuss the logarithmic concavity of the coefficients of the chromatic quasisymmetric function.

Cited in 3 Reviews
Cited in 13 Documents

MSC:

05E05 Symmetric functions and generalizations
05C15 Coloring of graphs and hypergraphs
05E10 Combinatorial aspects of representation theory
05A15 Exact enumeration problems, generating functions
05A30 \(q\)-calculus and related topics
05A17 Combinatorial aspects of partitions of integers
14M15 Grassmannians, Schubert varieties, flag manifolds
16T05 Hopf algebras and their applications

Keywords:

chromatic symmetric functions; LLT polynomials; \(q\)-hit numbers; planar networks; Hopf algebras
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References:

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