scholarly journals A Counterexample to a Conjecture on Schur Positivity of Chromatic Symmetric Functions of trees

10.37236/9696 ◽  
2020 ◽  
Vol 27 (4) ◽  
Author(s):  
Emmanuella Sandratra Rambeloson ◽  
John Shareshian
Keyword(s):  
Symmetric Function ◽  
Maximum Degree ◽  
Symmetric Functions ◽  
Chromatic Symmetric Function ◽  
Schur Positive

We show that no tree on twenty vertices with maximum degree ten has Schur positive chromatic symmetric function, thereby providing a counterexample to a conjecture of Dahlberg, She and van Willigenburg.

scholarly journals Power Sum Expansion of Chromatic Quasisymmetric Functions

10.37236/4761 ◽  
2015 ◽  
Vol 22 (2) ◽  
Author(s):  
Christos A. Athanasiadis
Keyword(s):  
Symmetric Group ◽  
Symmetric Function ◽  
Symmetric Functions ◽  
Unit Interval ◽  
Quasisymmetric Function ◽  
Interval Order ◽  
Combinatorial Formula ◽  
Natural Unit ◽  
Power Sum ◽  
Chromatic Symmetric Function

The chromatic quasisymmetric function of a graph was introduced by Shareshian and Wachs as a refinement of Stanley's chromatic symmetric function. An explicit combinatorial formula, conjectured by Shareshian and Wachs, expressing the chromatic quasisymmetric function of the incomparability graph of a natural unit interval order in terms of power sum symmetric functions, is proven. The proof uses a formula of Roichman for the irreducible characters of the symmetric group.

scholarly journals Chromatic Symmetric Functions of Hypertrees

10.37236/5369 ◽  
2017 ◽  
Vol 24 (2) ◽  
Author(s):  
Jair Taylor
Keyword(s):  
Symmetric Function ◽  
Symmetric Functions ◽  
Combinatorial Interpretation ◽  
Quasisymmetric Functions ◽  
Chromatic Symmetric Function

The chromatic symmetric function $X_H$ of a hypergraph $H$ is the sum of all monomials corresponding to proper colorings of $H$. When $H$ is an ordinary graph, it is known that $X_H$ is positive in the fundamental quasisymmetric functions $F_S$, but this is not the case for general hypergraphs. We exhibit a class of hypergraphs $H$ — hypertrees with prime-sized edges — for which $X_H$ is $F$-positive, and give an explicit combinatorial interpretation for the $F$-coefficients of $X_H$.

scholarly journals Hopf Algebra of Building Sets

10.37236/2413 ◽  
2012 ◽  
Vol 19 (4) ◽  
Author(s):  
Vladimir Grujić ◽  
Tanja Stojadinović
Keyword(s):  
Hopf Algebra ◽  
Symmetric Function ◽  
Symmetric Functions ◽  
Complete Characterization ◽  
Canonical Morphism ◽  
Simple Graphs ◽  
Building Set ◽  
Chromatic Symmetric Function ◽  
Building Sets

The combinatorial Hopf algebra on building sets $BSet$ extends the chromatic Hopf algebra of simple graphs. The image of a building set under canonical morphism to quasi-symmetric functions is the chromatic symmetric function of the corresponding hypergraph. By passing from graphs to building sets, we construct a sequence of symmetric functions associated to a graph. From the generalized Dehn-Sommerville relations for the Hopf algebra $BSet$, we define a class of building sets called eulerian and show that eulerian building sets satisfy Bayer-Billera relations. We show the existence of the $\mathbf{c}\mathbf{d}-$index, the polynomial in two noncommutative variables associated to an eulerian building set. The complete characterization of eulerian building sets is given in terms of combinatorics of intersection posets of antichains of finite sets.

scholarly journals Schur-Positivity in a Square

10.37236/3796 ◽  
2014 ◽  
Vol 21 (3) ◽  
Author(s):  
Cristina Ballantine ◽  
Rosa Orellana
Keyword(s):  
Symmetric Function ◽  
Symmetric Functions ◽  
Difficult Problem ◽  
Positivity Criterion ◽  
Schur Positive

Determining if a symmetric function is Schur-positive is a prevalent and, in general, a notoriously difficult problem.  In this paper we study the Schur-positivity of a family of symmetric functions.  Given a partition $\nu$, we denote by $\nu^c$ its complement in a square partition $(m^m)$.   We conjecture a  Schur-positivity criterion  for symmetric functions of the form $s_{\mu'}s_{\mu^c}-s_{\nu'}s_{\nu^c}$, where $\nu$ is a partition of weight $|\mu|-1$ contained in $\mu$ and the complement of $\mu$ is taken in the same square partition as the complement of $\nu$. We prove the conjecture in many cases.

scholarly journals A Two Parameter Chromatic Symmetric Function

10.37236/940 ◽  
2007 ◽  
Vol 14 (1) ◽  
Author(s):  
Ellison-Anne Williams
Keyword(s):  
Complete Graph ◽  
Symmetric Function ◽  
Symmetric Functions ◽  
Rational Functions ◽  
Macdonald Polynomials ◽  
Simple Graph ◽  
Macdonald Polynomial ◽  
Chromatic Symmetric Function ◽  
Correlation Properties ◽  
Two Parameter

We introduce and develop a two-parameter chromatic symmetric function for a simple graph $G$ over the field of rational functions in $q$ and $t,\,{\Bbb Q}(q,t)$. We derive its expansion in terms of the monomial symmetric functions, $m_{\lambda}$, and present various correlation properties which exist between the two-parameter chromatic symmetric function and its corresponding graph. Additionally, for the complete graph $G$ of order $n$, its corresponding two-parameter chromatic symmetric function is the Macdonald polynomial $Q_{(n)}$. Using this, we develop graphical analogues for the expansion formulas of the two-row Macdonald polynomials and the two-row Jack symmetric functions. Finally, we introduce the "complement" of this new function and explore some of its properties.

scholarly journals Schur and $e$-Positivity of Trees and Cut Vertices

10.37236/8930 ◽  
2020 ◽  
Vol 27 (1) ◽  
Author(s):  
Samantha Dahlberg ◽  
Adrian She ◽  
Stephanie Van Willigenburg
Keyword(s):  
Linear Combination ◽  
Connected Graph ◽  
Symmetric Function ◽  
Symmetric Functions ◽  
Perfect Matching ◽  
Schur Functions ◽  
Connected Components ◽  
Elementary Symmetric Functions ◽  
Chromatic Symmetric Function ◽  
Positive Linear Combination

We prove that the chromatic symmetric function of any $n$-vertex tree containing a vertex of degree $d\geqslant \log _2n +1$ is not $e$-positive, that is, not a positive linear combination of elementary symmetric functions. Generalizing this, we also prove that the chromatic symmetric function of any $n$-vertex connected graph containing a cut vertex whose deletion disconnects the graph into $d\geqslant\log _2n +1$ connected components is not $e$-positive. Furthermore we prove that any $n$-vertex bipartite graph, including all trees, containing a vertex of degree greater than $\lceil \frac{n}{2}\rceil$ is not Schur-positive, namely not a positive linear combination of Schur functions. In complete generality, we prove that if an $n$-vertex connected graph has no perfect matching (if $n$ is even) or no almost perfect matching (if $n$ is odd), then it is not $e$-positive. We hence deduce that many graphs containing the claw are not $e$-positive.

scholarly journals A Vertex-Weighted Tutte Symmetric Function, and Constructing Graphs with Equal Chromatic Symmetric Function

2021 ◽  
Vol 28 (2) ◽  
Author(s):  
José Aliste-Prieto ◽  
Logan Crew ◽  
Sophie Spirkl ◽  
José Zamora
Keyword(s):  
Symmetric Function ◽  
Spanning Tree ◽  
Symmetric Functions ◽  
Tutte Polynomial ◽  
Weighted Version ◽  
Spanning Forest ◽  
Chromatic Symmetric Function

This paper has two main parts. First, we consider the Tutte symmetric function XB, a generalization of the chromatic symmetric function. We introduce a vertex-weighted version of XB and show that this function admits a deletion-contraction relation. We also demonstrate that the vertex-weighted XB admits spanning-tree and spanning-forest expansions generalizing those of the Tutte polynomial by connecting XB to other graph functions. Second, we give several methods for constructing nonisomorphic graphs with equal chromatic and Tutte symmetric functions, and use them to provide specific examples.

scholarly journals How to get the weak order out of a digraph ?

2015 ◽  
Vol DMTCS Proceedings, 27th... (Proceedings) ◽  
Author(s):  
Francois Viard
Keyword(s):  
Wreath Product ◽  
Symmetric Function ◽  
Coxeter Groups ◽  
Symmetric Functions ◽  
Weak Order ◽  
Möbius Function ◽  
Mobius Function ◽  
Acyclic Digraph ◽  
International Audience

International audience We construct a poset from a simple acyclic digraph together with a valuation on its vertices, and we compute the values of its Möbius function. We show that the weak order on Coxeter groups $A$<sub>$n-1$</sub>, $B$<sub>$n$</sub>, $Ã$<sub>$n$</sub>, and the flag weak order on the wreath product &#8484;<sub>$r$</sub> &#8768; $S$<sub>$n$</sub> introduced by Adin, Brenti and Roichman (2012), are special instances of our construction. We conclude by briefly explaining how to use our work to define quasi-symmetric functions, with a special emphasis on the $A$<sub>$n-1$</sub> case, in which case we obtain the classical Stanley symmetric function. On construit une famille d’ensembles ordonnés à partir d’un graphe orienté, simple et acyclique munit d’une valuation sur ses sommets, puis on calcule les valeurs de leur fonction de Möbius respective. On montre que l’ordre faible sur les groupes de Coxeter $A$<sub>$n-1$</sub>, $B$<sub>$n$</sub>, $Ã$<sub>$n$</sub>, ainsi qu’une variante de l’ordre faible sur les produits en couronne &#8484;<sub>$r$</sub> &#8768; $S$<sub>$n$</sub> introduit par Adin, Brenti et Roichman (2012), sont des cas particuliers de cette construction. On conclura en expliquant brièvement comment ce travail peut-être utilisé pour définir des fonction quasi-symétriques, en insistant sur l’exemple de l’ordre faible sur $A$<sub>$n-1$</sub> où l’on obtient les séries de Stanley classiques.

scholarly journals A Quasisymmetric Function Generalization of the Chromatic Symmetric Function

10.37236/518 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Brandon Humpert
Keyword(s):  
Symmetric Function ◽  
Chromatic Polynomial ◽  
Quasisymmetric Function ◽  
Quasisymmetric Functions ◽  
Acyclic Orientations ◽  
Chromatic Symmetric Function

The chromatic symmetric function $X_G$ of a graph $G$ was introduced by Stanley. In this paper we introduce a quasisymmetric generalization $X^k_G$ called the $k$-chromatic quasisymmetric function of $G$ and show that it is positive in the fundamental basis for the quasisymmetric functions. Following the specialization of $X_G$ to $\chi_G(\lambda)$, the chromatic polynomial, we also define a generalization $\chi^k_G(\lambda)$ and show that evaluations of this polynomial for negative values generalize a theorem of Stanley relating acyclic orientations to the chromatic polynomial.

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