scholarly journals LLT polynomials, chromatic quasisymmetric functions and graphs with cycles

2018 ◽  
Vol 341 (12) ◽  
pp. 3453-3482 ◽  
Author(s):  
Per Alexandersson ◽  
Greta Panova
Keyword(s):  
Quasisymmetric Functions ◽  
Llt Polynomials ◽  
Functions And Graphs

scholarly journals P-Partitions and p-Positivity

2019 ◽  
Author(s):  
Per Alexandersson ◽  
Robin Sulzgruber
Keyword(s):  
Linear Combination ◽  
Generating Functions ◽  
Quasisymmetric Functions ◽  
Vertical Strip ◽  
Power Sums ◽  
Llt Polynomials ◽  
Tutte Polynomials

AbstractUsing the combinatorics of $\alpha$-unimodal sets, we establish two new results in the theory of quasisymmetric functions. First, we obtain the expansion of the fundamental basis into quasisymmetric power sums. Secondly, we prove that generating functions of reverse $P$-partitions expand positively into quasisymmetric power sums. Consequently, any nonnegative linear combination of such functions is $p$-positive whenever it is symmetric. As an application, we derive positivity results for chromatic quasisymmetric functions, unicellular and vertical strip LLT polynomials, multivariate Tutte polynomials, and the more general $B$-polynomials, matroid quasisymmetric functions, and certain Eulerian quasisymmetric functions, thus reproving and improving on numerous results in the literature.

scholarly journals Dual Creation Operators and a Dendriform Algebra Structure on the Quasisymmetric Functions

2017 ◽  
Vol 69 (1) ◽  
pp. 21-53 ◽  
Author(s):  
Darij Grinberg
Keyword(s):  
Hopf Algebras ◽  
Young Tableau ◽  
Vertex Operators ◽  
Schur Function ◽  
Algebra Structure ◽  
Quasisymmetric Functions ◽  
Combinatorial Hopf Algebras ◽  
Dendriform Algebra ◽  
Natural Analogues ◽  
Creation Operators

AbstractThe dual immaculate functions are a basis of the ring QSym of quasisymmetric functions and form one of the most natural analogues of the Schur functions. The dual immaculate function corresponding to a composition is a weighted generating function for immaculate tableaux in the same way as a Schur function is for semistandard Young tableaux; an immaculate tableau is defined similarly to a semistandard Young tableau, but the shape is a composition rather than a partition, and only the first column is required to strictly increase (whereas the other columns can be arbitrary, but each row has to weakly increase). Dual immaculate functions were introduced by Berg, Bergeron, Saliola, Serrano, and Zabrocki in arXiv:1208.5191, and have since been found to possess numerous nontrivial properties.In this note, we prove a conjecture of M. Zabrocki that provides an alternative construction for the dual immaculate functions in terms of certain “vertex operators”. The proof uses a dendriform structure on the ring QSym; we discuss the relation of this structure to known dendriformstructures on the combinatorial Hopf algebras FQSym andWQSym.

scholarly journals A Note on Three Types of Quasisymmetric Functions

10.37236/1958 ◽  
2005 ◽  
Vol 12 (1) ◽  
Author(s):  
T. Kyle Petersen
Keyword(s):  
Generating Functions ◽  
Type A ◽  
Type B ◽  
Quasisymmetric Functions ◽  
Descent Algebra ◽  
Combinatorial Description ◽  
Partition Theorems

In the context of generating functions for $P$-partitions, we revisit three flavors of quasisymmetric functions: Gessel's quasisymmetric functions, Chow's type B quasisymmetric functions, and Poirier's signed quasisymmetric functions. In each case we use the inner coproduct to give a combinatorial description (counting pairs of permutations) to the multiplication in: Solomon's type A descent algebra, Solomon's type B descent algebra, and the Mantaci-Reutenauer algebra, respectively. The presentation is brief and elementary, our main results coming as consequences of $P$-partition theorems already in the literature.

scholarly journals Colored Trees and Noncommutative Symmetric Functions

10.37236/468 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Matt Szczesny
Keyword(s):  
Hopf Algebra ◽  
Symmetric Functions ◽  
Hall Algebra ◽  
Quasisymmetric Functions ◽  
Noncommutative Symmetric Functions

Let ${\cal CRF}_S$ denote the category of $S$-colored rooted forests, and H$_{{\cal CRF}_S}$ denote its Ringel-Hall algebra as introduced by Kremnizer and Szczesny. We construct a homomorphism from a $K^+_0({\cal CRF}_S)$–graded version of the Hopf algebra of noncommutative symmetric functions to H$_{{\cal CRF}_S}$. Dualizing, we obtain a homomorphism from the Connes-Kreimer Hopf algebra to a $K^+_0({\cal CRF}_S)$–graded version of the algebra of quasisymmetric functions. This homomorphism is a refinement of one considered by W. Zhao.

scholarly journals Antipode Formulas for some Combinatorial Hopf Algebras

10.37236/5949 ◽  
2016 ◽  
Vol 23 (4) ◽  
Author(s):  
Rebecca Patrias
Keyword(s):  
Hopf Algebra ◽  
Hopf Algebras ◽  
Symmetric Functions ◽  
Explicit Formulas ◽  
Quasisymmetric Functions ◽  
Noncommutative Symmetric Functions ◽  
Combinatorial Hopf Algebras ◽  
Combinatorial Formulas ◽  
K Theory

Motivated by work of Buch on set-valued tableaux in relation to the K-theory of the Grassmannian, Lam and Pylyavskyy studied six combinatorial Hopf algebras that can be thought of as K-theoretic analogues of the Hopf algebras of symmetric functions, quasisymmetric functions, noncommutative symmetric functions, and of the Malvenuto-Reutenauer Hopf algebra of permutations. They described the bialgebra structure in all cases that were not yet known but left open the question of finding explicit formulas for the antipode maps. We give combinatorial formulas for the antipode map for the K-theoretic analogues of the symmetric functions, quasisymmetric functions, and noncommutative symmetric functions.

scholarly journals Schubert Polynomials and $k$-Schur Functions

10.37236/4139 ◽  
2014 ◽  
Vol 21 (4) ◽  
Author(s):  
Carolina Benedetti ◽  
Nantel Bergeron
Keyword(s):  
Finite Type ◽  
Schur Functions ◽  
Type A ◽  
Bruhat Order ◽  
Schur Function ◽  
Quasisymmetric Functions ◽  
Affine Grassmannian ◽  
Schubert Polynomial ◽  
Schubert Polynomials ◽  
General Program

The main purpose of this paper is to show that the multiplication of a Schubert polynomial of finite type $A$ by a Schur function, which we refer to as Schubert vs. Schur problem, can be understood combinatorially from the multiplication in the space of dual $k$-Schur functions. Using earlier work by the second author, we encode both problems by means of quasisymmetric functions. On the Schubert vs. Schur side, we study the poset given by the Bergeron-Sottile's $r$-Bruhat order, along with certain operators associated to this order. Then, we connect this poset with a graph on dual $k$-Schur functions given by studying the affine grassmannian order of  Lam-Lapointe-Morse-Shimozono. Also, we define operators associated to the graph on dual $k$-Schur functions which are analogous to the ones given for the Schubert vs. Schur problem. This is the first step of our more general program of showing combinatorially  the positivity of the multiplication of a dual $k$-Schur function by a Schur function.

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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