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 On the Schur Expansion of Hall-Littlewood and Related Polynomials via Yamanouchi Words

10.37236/6732 ◽  
2017 ◽  
Vol 24 (1) ◽  
Author(s):  
Austin Roberts
Keyword(s):  
Symmetric Function ◽  
Symmetric Functions ◽  
Macdonald Polynomials ◽  
Macdonald Polynomial ◽  
Gamma Delta ◽  
Colored Graph ◽  
Littlewood Polynomials ◽  
Dual Equivalence ◽  
Leading Term ◽  
Definition Of

This paper uses the theory of dual equivalence graphs to give explicit Schur expansions for several families of symmetric functions. We begin by giving a combinatorial definition of the modified Macdonald polynomials and modified Hall-Littlewood polynomials indexed by any diagram $\delta \subset {\mathbb Z} \times {\mathbb Z}$, written as $\widetilde H_{\delta}(X;q,t)$ and $\widetilde H_{\delta}(X;0,t)$, respectively. We then give an explicit Schur expansion of $\widetilde H_{\delta}(X;0,t)$ as a sum over a subset of the Yamanouchi words, as opposed to the expansion using the charge statistic given in 1978 by Lascoux and Schüztenberger. We further define the symmetric function $R_{\gamma,\delta}(X)$ as a refinement of $\widetilde H_{\delta}(X;0,t)$ and similarly describe its Schur expansion. We then analyze $R_{\gamma,\delta}(X)$ to determine the leading term of its Schur expansion. We also provide a conjecture towards the Schur expansion of $\widetilde H_{\delta}(X;q,t)$. To gain these results, we use a construction from the 2007 work of Sami Assaf to associate each Macdonald polynomial with a signed colored graph $\mathcal{H}_\delta$. In the case where a subgraph of $\mathcal{H}_\delta$ is a dual equivalence graph, we provide the Schur expansion of its associated symmetric function, yielding several corollaries.

scholarly journals Hall-Littlewood Polynomials in terms of Yamanouchi words

2014 ◽  
Vol DMTCS Proceedings vol. AT,... (Proceedings) ◽  
Author(s):  
Austin Roberts
Keyword(s):  
Symmetric Function ◽  
Symmetric Functions ◽  
Macdonald Polynomials ◽  
Macdonald Polynomial ◽  
Colored Graph ◽  
Littlewood Polynomials ◽  
Dual Equivalence ◽  
International Audience ◽  
Leading Term ◽  
Definition Of

International audience This paper uses the theory of dual equivalence graphs to give explicit Schur expansions to several families of symmetric functions. We begin by giving a combinatorial definition of the modified Macdonald polynomials and modified Hall-Littlewood polynomials indexed by any diagram $δ ⊂ \mathbb{Z} \times \mathbb{Z}$, written as $\widetilde H_δ (X;q,t)$ and $\widetilde P_δ (X;t)$, respectively. We then give an explicit Schur expansion of $\widetilde P_δ (X;t)$ as a sum over a subset of the Yamanouchi words, as opposed to the expansion using the charge statistic given in 1978 by Lascoux and Schüztenberger. We further define the symmetric function $R_γ ,δ (X)$ as a refinement of $\widetilde P_δ$ and similarly describe its Schur expansion. We then analysize $R_γ ,δ (X)$ to determine the leading term of its Schur expansion. To gain these results, we associate each Macdonald polynomial with a signed colored graph $\mathcal{H}_δ$ . In the case where a subgraph of $\mathcal{H}_δ$ is a dual equivalence graph, we provide the Schur expansion of its associated symmetric function, yielding several corollaries.

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 A Macdonald Vertex Operator and Standard Tableaux Statistics

10.37236/1383 ◽  
1998 ◽  
Vol 5 (1) ◽  
Author(s):  
Mike Zabrocki
Keyword(s):  
Vertex Operator ◽  
Generating Functions ◽  
Symmetric Function ◽  
Symmetric Functions ◽  
Schur Function ◽  
Inductive Proof ◽  
Integer Coefficients ◽  
The Family ◽  
The Relationship ◽  
Two Parameter

The two parameter family of coefficients $K_{\lambda \mu}(q,t)$ introduced by Macdonald are conjectured to $(q,t)$ count the standard tableaux of shape $\lambda $. If this conjecture is correct, then there exist statistics $a_\mu(T)$ and $b_\mu(T)$ such that the family of symmetric functions $H_\mu[X;q,t] = \sum_\lambda K_{\lambda \mu}(q,t) s_\lambda [X]$ are generating functions for the standard tableaux of size $|\mu|$ in the sense that $H_\mu[X;q,t] = \sum_{T} q^{a_\mu(T)} t^{b_\mu(T)} s_{\lambda (T)}[X]$ where the sum is over standard tableau of of size $|\mu|$. We give a formula for a symmetric function operator $H_2^{qt}$ with the property that $H_2^{qt} H_{(2^a1^b)}[X;q,t]= H_{(2^{a+1}1^b)}[X;q,t]$. This operator has a combinatorial action on the Schur function basis. We use this Schur function action to show by induction that $H_{(2^a1^b)}[X;q,t]$ is the generating function for standard tableaux of size $2a+b$ (and hence that $K_{\lambda (2^a1^b)}(q,t)$ is a polynomial with non-negative integer coefficients). The inductive proof gives an algorithm for 'building' the standard tableaux of size $n+2$ from the standard tableaux of size $n$ and divides the standard tableaux into classes that are generalizations of the catabolism type. We show that reversing this construction gives the statistics $a_\mu(T)$ and $b_\mu(T)$ when $\mu$ is of the form $(2^a1^b)$ and that these statistics prove conjectures about the relationship between adjacent rows of the $(q,t)$-Kostka matrix that were suggested by Lynne Butler.

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 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 Staircase Macdonald polynomials and the $q$-Discriminant

2008 ◽  
Vol DMTCS Proceedings vol. AJ,... (Proceedings) ◽  
Author(s):  
Adrien Boussicault ◽  
Jean-Gabriel Luque
Keyword(s):  
Symmetric Functions ◽  
Macdonald Polynomials ◽  
Macdonald Polynomial ◽  
Monomial Basis ◽  
Nous Montrons ◽  
International Audience ◽  
Nous Examinons

International audience We prove that a $q$-deformation $\mathfrak{D}_k(\mathbb{X};q)$ of the powers of the discriminant is equal, up to a normalization, to a specialization of a Macdonald polynomial indexed by a staircase partition. We investigate the expansion of $\mathfrak{D}_k(\mathbb{X};q)$ on different bases of symmetric functions. In particular, we show that its expansion on the monomial basis can be explicitly described in terms of standard tableaux and we generalize a result of King-Toumazet-Wybourne about the expansion of the $q$-discriminant on the Schur basis. Nous montrons qu’une $q$-déformation $\mathfrak{D}_k(\mathbb{X};q)$ des puissances du discriminant est égale, à un coefficient de normalisation près, à un polynôme de Macdonald indexé par une partition escalier pour une certaine spécialisation des paramètres. Nous examinons les développements de $\mathfrak{D}_k(\mathbb{X};q)$ dans différentes bases de fonctions symétriques. En particulier, nous montrons que son écriture dans la base des fonctions monomiales peut être explicitement décrite en terme de tableaux standard et nous généralisons un résultat de King-Toumazet-Wybourne sur le développement du $q$-discriminant dans la base de Schur.

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.

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