LLT polynomials, chromatic quasisymmetric functions and graphs with cycles

Alexandersson, Per; Panova, Greta

doi:10.1016/j.disc.2018.09.001

Cited by 36 publications

(69 citation statements)

References 19 publications

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“…As we already mentioned in the introduction, our result implies several e-positivity conjectured in [AP18], [Ber17] and [GHQR19]. For example, we have the following corollary.…”

Section: Some Consequencessupporting

confidence: 84%

“…As a corollary of our main result, we get many of the e-positivity conjectured in [Ber17], [AP18] and [GHQR19]. But we want to stress here that not all the e-positivities observed in [Ber17], [AP18] and [GHQR19] follow from our results. Moreover, in [Ale19] and [GHQR19] the authors conjecture explicit formulas for the expansion in the elementary symmetric functions basis of G ν [X; q +1] when G ν [X; q] is a vertical strip LLT polynomial.…”

Section: Introductionsupporting

confidence: 58%

“…Many of these conjectures would follow easily if that same property were true for vertical strip LLT polynomials. This last property has been conjectured both in [AP18] and [GHQR19], and it was observed in [AP18] that it has an interesting parallel with a famous e-positivity conjecture about the chromatic symmetric functions introduced by Shareshian and Wachs in [SW12]. Notice also that in [AP18], [GHQR19] and [Ale19] some special cases have been proved.…”

Section: Introductionmentioning

confidence: 76%

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e-Positivity of vertical strip LLT polynomials

D'Adderio

2020

Journal of Combinatorial Theory, Series A

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In this article we prove the e-positivity of Gν [X; q + 1] when Gν[X; q] is a vertical strip LLT polynomial. This property has been conjectured in [AP18] and [GHQR19], and it implies several e-positivities conjectured in those references and in [Ber17].We make use of a result of Carlsson and Mellit [CM18] that shows that a vertical strip LLT polynomial can be obtained by applying certain compositions of operators of the Dyck path algebra to the constant 1. Our proof gives in fact an algorithm to expand these symmetric functions in the elementary basis, and it shows, as a byproduct, that these compositions of operators are actually multiplication operators.

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“…As we already mentioned in the introduction, our result implies several e-positivity conjectured in [AP18], [Ber17] and [GHQR19]. For example, we have the following corollary.…”

Section: Some Consequencessupporting

confidence: 84%

Section: Introductionsupporting

confidence: 58%

Section: Introductionmentioning

confidence: 76%

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e-Positivity of vertical strip LLT polynomials

D'Adderio

2020

Journal of Combinatorial Theory, Series A

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“…We use the same notation and terminology as in [AP18]. The reader is assumed to have a basic background on symmetric functions and related combinatorial objects, see [Sta01,Mac95].…”

Section: Preliminariesmentioning

confidence: 99%

“…Example 2. We can illustrate area sequences and their corresponding unit-interval graphs as Dyck diagrams, as is done in [Hag07,AP18]. For example, (0, 1, 2, 3, 2, 2) corresponds to the diagram where the area sequence specify the number of white squares in each row, bottom to top.…”

Section: Dyck Paths and Unit-interval Graphsmentioning

confidence: 99%

LLT polynomials, elementary symmetric functions and melting lollipops

Alexandersson

2020

J Algebr Comb

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We conjecture an explicit positive combinatorial formula for the expansion of unicellular LLT polynomials in the elementary symmetric basis. This is an analogue of the Shareshian-Wachs conjecture previously studied by Panova and the author in 2018. We show that the conjecture for unicellular LLT polynomials implies a similar formula for vertical-strip LLT polynomials.We prove positivity in the elementary basis in for the class of graphs called "melting lollipops" previously considered by Huh, Nam and Yoo. This is done by proving a curious relationship between a generalization of charge and orientations of unit-interval graphs.We also provide short bijective proofs of Lee's three-term recurrences for unicellular LLT polynomials and we show that these recurrences are enough to generate all unicellular LLT polynomials associated with abelian area sequences.

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Recent Trends in Quasisymmetric Functions

Mason

2019

Association for Women in Mathematics Series

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This article serves as an introduction to several recent developments in the study of quasisymmetric functions. The focus of this survey is on connections between quasisymmetric functions and the combinatorial Hopf algebra of noncommutative symmetric functions, appearances of quasisymmetric functions within the theory of Macdonald polynomials, and analogues of symmetric functions. Topics include the significance of quasisymmetric functions in representation theory (such as representations of the 0-Hecke algebra), recently discovered bases (including analogues of well-studied symmetric function bases), and applications to open problems in symmetric function theory.

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LLT polynomials, chromatic quasisymmetric functions and graphs with cycles

Cited by 36 publications

References 19 publications

e-Positivity of vertical strip LLT polynomials

e-Positivity of vertical strip LLT polynomials

LLT polynomials, elementary symmetric functions and melting lollipops

Recent Trends in Quasisymmetric Functions

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