Power Sum Expansion of Chromatic Quasisymmetric Functions

Athanasiadis, Christos A.

doi:10.37236/4761

Cited by 25 publications

(47 citation statements)

References 10 publications

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“…The unicellular case is a straightforward consequence of Lemma 12 (see [AP18,HW17]) together with the power-sum expansion formula for the chromatic symmetric symmetric functions. We note that the formula in the chromatic case was first conjectured by Shareshian-Wachs and later proved by Athanasiadis [Ath15].…”

Section: A Possible Approach To Settle the Main Conjecturementioning

confidence: 57%

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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Section: A Possible Approach To Settle the Main Conjecturementioning

confidence: 57%

LLT polynomials, elementary symmetric functions and melting lollipops

Alexandersson

2020

J Algebr Comb

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“…Sections [5][6][7][8] show that nearly all examples of fine S n -characters mentioned earlier have natural B n -analogues. These include the irreducible characters of B n , the Gelfand model, the characters of the natural B n -action on the homogeneous components of the coinvariant algebra of type B, signed analogues of the Lie characters, characters induced from exterior algebras and k-root enumerators, with corresponding fine sets consisting of elements of Knuth classes of type B n , involutions, signed permutations of fixed flaginversion number or flag-major index, conjugacy classes in B n , signed analogues of arc permutations and k-roots of the identity signed permutation, respectively.…”

Section: Introductionmentioning

confidence: 86%

“…The concept of α-unimodality, and Theorem 2.4 in particular, were recently applied to prove conjectures of Regev concerning induced characters [17, Section 9] and of Shareshian and Wachs concerning chromatic quasisymmetric functions [8].…”

Section: Characters and Symmetric Functionsmentioning

confidence: 99%

Character formulas and descents for the hyperoctahedral group

Adin

Athanasiadis

Elizalde

et al. 2017

Advances in Applied Mathematics

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A general setting to study a certain type of formulas, expressing characters of the symmetric group S n explicitly in terms of descent sets of combinatorial objects, has been developed by two of the authors. This theory is further investigated in this paper and extended to the hyperoctahedral group B n . Key ingredients are a new formula for the irreducible characters of B n , the signed quasisymmetric functions introduced by Poirier, and a new family of matrices of Walsh-Hadamard type. Applications include formulas for natural B n -actions on coinvariant and exterior algebras and on the top homology of a certain poset in terms of the combinatorics of various classes of signed permutations, as well as a B n -analogue of an equidistribution theorem of Désarménien and Wachs.

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“…Let jdt(k + T ) be the resulting standard tableau. In Lemmas 4.5 and 4.7 we will show that jdt is a Des-preserving bijection as described in Equation (5). First, we introduce some terminology that will be used in the proofs.…”

Section: In Particularmentioning

confidence: 99%

“…The problem of determining whether a given subset of permutations is symmetric and Schur-positive was first posed in [11], see also [17], [10] and [19]. The search for Schur-positive subsets is an active area of research [2,16,15,5,1,9].…”

Section: Introductionmentioning

confidence: 99%

On rotated Schur-positive sets

Elizalde

Roichman

2017

Journal of Combinatorial Theory, Series A

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The problem of finding Schur-positive sets of permutations, originally posed by Gessel and Reutenauer, has seen some recent developments. Schur-positive sets of pattern-avoiding permutations have been found by Sagan et al and a general construction based on geometric operations on grid classes has been given by the authors. In this paper we prove that horizontal rotations of Schur-positive subsets of permutations are always Schur-positive. The proof applies a cyclic action on standard Young tableaux of certain skew shapes and a jeu-de-taquin type straightening algorithm. As a consequence of the proof we obtain a notion of cyclic descent set on these tableaux, which is rotated by the cyclic action on them.

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Power Sum Expansion of Chromatic Quasisymmetric Functions

Cited by 25 publications

References 10 publications

LLT polynomials, elementary symmetric functions and melting lollipops

LLT polynomials, elementary symmetric functions and melting lollipops

Character formulas and descents for the hyperoctahedral group

On rotated Schur-positive sets

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