scholarly journals Flow Polytopes and the Space of Diagonal Harmonics

2019 ◽  
Vol 71 (6) ◽  
pp. 1495-1521
Author(s):  
Ricky Ini Liu ◽  
Alejandro H. Morales ◽  
Karola Mészáros
Keyword(s):  
Complete Graph ◽  
Hilbert Series ◽  
Threshold Graphs ◽  
Flow Polytopes ◽  
Diagonal Harmonics ◽  
Flow Polytope

AbstractA result of Haglund implies that the$(q,t)$-bigraded Hilbert series of the space of diagonal harmonics is a$(q,t)$-Ehrhart function of the flow polytope of a complete graph with netflow vector$(-n,1,\ldots ,1)$. We study the$(q,t)$-Ehrhart functions of flow polytopes of threshold graphs with arbitrary netflow vectors. Our results generalize previously known specializations of the mentioned bigraded Hilbert series at$t=1$,$0$, and$q^{-1}$. As a corollary to our results, we obtain a proof of a conjecture of Armstrong, Garsia, Haglund, Rhoades, and Sagan about the$(q,q^{-1})$-Ehrhart function of the flow polytope of a complete graph with an arbitrary netflow vector.

scholarly journals Q-borderenergetic threshold graphs

2020 ◽  
Vol 42 ◽  
pp. e91
Author(s):  
João Roberto Lazzarin ◽  
Oscar Franscisco Másquez Sosa ◽  
Fernando Colman Tura
Keyword(s):  
Complete Graph ◽  
Threshold Graphs ◽  
Laplacian Energy ◽  
Signless Laplacian

A graph G is said to be borderenergetic (L-borderenergetic, respectively) if its energy (Laplacian energy, respectively) equals the energy (Laplacian energy, respectively) of the complete graph. Recently, this concept was extend to signless Laplacian energy (see Tao, Q., Hou, Y. (2018). Q-borderenergetic graphs. AKCE International Journal of Graphs and Combinatorics). A graph G is called Q-borderenergetic if its signless Laplacian energy is the same of the complete graph Kn; i.e., QE(G) = QE(Kn) = 2n - 2: In this paper, we investigate Q-borderenergetic graphs on the class of threshold graphs. For a family of threshold graphs of order n = 100; we find out exactly 13 graphs such that QE(G) = 2n- 2:

scholarly journals Generalized Tesler matrices, virtual Hilbert series, and Macdonald polynomial operators

2015 ◽  
Vol DMTCS Proceedings, 27th... (Proceedings) ◽  
Author(s):  
Andrew Timothy Wilson
Keyword(s):  
Symmetric Function ◽  
Hilbert Series ◽  
Inner Product ◽  
Binomial Coefficients ◽  
Diagonal Harmonics ◽  
Ordered Set Partitions ◽  
International Audience ◽  
Tesler Matrices ◽  
Matrix Expression ◽  
Polynomial Operators

International audience We generalize previous definitions of Tesler matrices to allow negative matrix entries and non-positive hook sums. Our main result is an algebraic interpretation of a certain weighted sum over these matrices. Our interpretation uses <i>virtual Hilbert series</i>, a new class of symmetric function specializations which are defined by their values on (modified) Macdonald polynomials. As a result of this interpretation, we obtain a Tesler matrix expression for the Hall inner product $\langle \Delta_f e_n, p_{1^{n}}\rangle$, where $\Delta_f$ is a symmetric function operator from the theory of diagonal harmonics. We use our Tesler matrix expression, along with various facts about Tesler matrices, to provide simple formulas for $\langle \Delta_{e_1} e_n, p_{1^{n}}\rangle$ and $\langle \Delta_{e_k} e_n, p_{1^{n}}\rangle \mid_{t=0}$ involving $q; t$-binomial coefficients and ordered set partitions, respectively. Nous généralisons les définitions précédentes de matrices Tesler pour permettre les entrées de la matrice négatives et des montants crochet non-positifs. Notre principal résultat est une interprétation algébrique d’une certaine somme pondérée sur ces matrices. Notre interprétation utilise <i>série de Hilbert virtuel</i>, une nouvelle classe de spécialisations fonctionnelles symétriques qui sont définies par leurs valeurs sur les polynômes de Macdonald (modifiées). À la suite de cette interprétation, on obtient une expression de la matrice Tesler pour la salle intérieure produit $\langle \Delta_f e_n, p_{1^{n}}\rangle$, où $\Delta_f$ est un opérateur de fonction symétrique de la théorie harmonique de diagonale. Nous utilisons notre expression de la matrice Tesler, ainsi que divers faits sur des matrices Tesler, de fournir des formules simples pour $\langle \Delta_{e_1} e_n, p_{1^{n}}\rangle$ et $\langle \Delta_{e_k} e_n, p_{1^{n}}\rangle \mid_{t=0}$ impliquant $q; t$-coefficients binomial et ensemble ordonné partitions, respectivement.

scholarly journals Special Cases of the Parking Functions Conjecture and Upper-Triangular Matrices

2011 ◽  
Vol DMTCS Proceedings vol. AO,... (Proceedings) ◽  
Author(s):  
Paul Levande
Keyword(s):  
Generating Function ◽  
Hilbert Series ◽  
Parking Functions ◽  
Combinatorial Interpretation ◽  
Forthcoming Article ◽  
Triangular Matrices ◽  
Upper Triangular Matrices ◽  
Diagonal Harmonics ◽  
Special Cases ◽  
Bivariate Generating Function

International audience We examine the $q=1$ and $t=0$ special cases of the parking functions conjecture. The parking functions conjecture states that the Hilbert series for the space of diagonal harmonics is equal to the bivariate generating function of $area$ and $dinv$ over the set of parking functions. Haglund recently proved that the Hilbert series for the space of diagonal harmonics is equal to a bivariate generating function over the set of Tesler matrices–upper-triangular matrices with every hook sum equal to one. We give a combinatorial interpretation of the Haglund generating function at $q=1$ and prove the corresponding case of the parking functions conjecture (first proven by Garsia and Haiman). We also discuss a possible proof of the $t = 0$ case consistent with this combinatorial interpretation. We conclude by briefly discussing possible refinements of the parking functions conjecture arising from this research and point of view. $\textbf{Note added in proof}$: We have since found such a proof of the $t = 0$ case and conjectured more detailed refinements. This research will most likely be presented in full in a forthcoming article. On examine les cas spéciaux $q=1$ et $t=0$ de la conjecture des fonctions de stationnement. Cette conjecture déclare que la série de Hilbert pour l'espace des harmoniques diagonaux est égale à la fonction génératrice bivariée (paramètres $area$ et $dinv$) sur l'ensemble des fonctions de stationnement. Haglund a prouvé récemment que la série de Hilbert pour l'espace des harmoniques diagonaux est égale à une fonction génératrice bivariée sur l'ensemble des matrices de Tesler triangulaires supérieures dont la somme de chaque équerre vaut un. On donne une interprétation combinatoire de la fonction génératrice de Haglund pour $q=1$ et on prouve le cas correspondant de la conjecture dans le cas des fonctions de stationnement (prouvé d'abord par Garsia et Haiman). On discute aussi d'une preuve possible du cas $t=0$, cohérente avec cette interprétation combinatoire. On conclut en discutant brièvement les raffinements possibles de la conjecture des fonctions de stationnement de ce point de vue. $\textbf{Note ajoutée sur épreuve}$: j'ai trouvé depuis cet article une preuve du cas $t=0$ et conjecturé des raffinements possibles. Ces résultats seront probablement présentés dans un article ultérieur.

scholarly journals Hyperplane Arrangements and Diagonal Harmonics

2011 ◽  
Vol DMTCS Proceedings vol. AO,... (Proceedings) ◽  
Author(s):  
Drew Armstrong
Keyword(s):  
Symmetric Functions ◽  
Hilbert Series ◽  
Hyperplane Arrangement ◽  
Type A ◽  
Hyperplane Arrangements ◽  
Combinatorial Interpretation ◽  
Nabla Operator ◽  
Affine Weyl Group ◽  
Diagonal Harmonics ◽  
International Audience

International audience In 2003, Haglund's bounce statistic gave the first combinatorial interpretation of the q,t-Catalan numbers and the Hilbert series of diagonal harmonics. In this paper we propose a new combinatorial interpretation in terms of the affine Weyl group of type A. In particular, we define two statistics on affine permutations; one in terms of the Shi hyperplane arrangement, and one in terms of a new arrangement — which we call the Ish arrangement. We prove that our statistics are equivalent to the area' and bounce statistics of Haglund and Loehr. In this setting, we observe that bounce is naturally expressed as a statistic on the root lattice. We extend our statistics in two directions: to "extended'' Shi arrangements and to the bounded chambers of these arrangements. This leads to a (conjectural) combinatorial interpretation for all integral powers of the Bergeron-Garsia nabla operator applied to elementary symmetric functions. En 2003, la statistique bounce de Haglund a donné la première interprétation combinatoire de la somme des nombres q,t-Catalan et de la série de Hilbert des harmoniques diagonaux. Dans cet article nous proposons une nouvelle interprétation combinatoire à partir du groupe de Weyl affine de type A. En particulier, nous définissons deux statistiques sur les permutations affines; l'une à partir de l'arrangement d'hyperplans Shi, et l'autre à partir d'un nouvel arrangement — que nous appelons l'arrangement Ish. Nous prouvons que nos statistiques sont équivalentes aux statistiques area' et bounce de Haglund et Loehr. Dans ce contexte, nous observons que bounce s'exprime naturellement comme une statistique sur le réseau des racines. Nous prolongeons nos statistiques dans deux directions: arrangements Shi "étendus'', et chambres bornées associées. Cela conduit à une interprétation (conjecturale) combinatoire pour toutes les puissances entières de l'opérateur nabla de Bergeron-Garsia appliqué aux fonctions symétriques élémentaires.

scholarly journals On Parking Functions and the Zeta Map in Types B, C and D

10.37236/6714 ◽  
2018 ◽  
Vol 25 (1) ◽  
Author(s):  
Robin Sulzgruber ◽  
Marko Thiel
Keyword(s):  
Hilbert Series ◽  
Root Systems ◽  
The Other ◽  
Lattice Paths ◽  
Square Lattice ◽  
Parking Functions ◽  
Combinatorial Objects ◽  
Coxeter Number ◽  
Dyck Paths ◽  
Diagonal Harmonics

Let $\Phi$ be an irreducible crystallographic root system with Weyl group $W$, coroot lattice $\check{Q}$ and Coxeter number $h$. Recently the second named author defined a uniform $W$-isomorphism $\zeta$ between the finite torus $\check{Q}/(mh+1)\check{Q}$ and the set of non-nesting parking functions $\operatorname{Park}^{(m)}(\Phi)$. If $\Phi$ is of type $A_{n-1}$ and $m=1$ this map is equivalent to a map defined on labelled Dyck paths that arises in the study of the Hilbert series of the space of diagonal harmonics. In this paper we investigate the case $m=1$ for the other infinite families of root systems ($B_n$, $C_n$ and $D_n$). In each type we define models for the finite torus and for the set of non-nesting parking functions in terms of labelled lattice paths. The map $\zeta$ can then be viewed as a map between these combinatorial objects. Our work entails new bijections between (square) lattice paths and ballot paths.

scholarly journals Conjectured Combinatorial Models for the Hilbert Series of Generalized Diagonal Harmonics Modules

10.37236/1821 ◽  
2004 ◽  
Vol 11 (1) ◽  
Author(s):  
Nicholas A. Loehr ◽  
Jeffrey B. Remmel
Keyword(s):  
Generating Function ◽  
Hilbert Series ◽  
Lattice Paths ◽  
Weighted Sums ◽  
Parking Functions ◽  
Dyck Paths ◽  
Diagonal Harmonics ◽  
Combinatorial Formulas ◽  
New Statistics ◽  
Univariate Distribution

Haglund and Loehr previously conjectured two equivalent combinatorial formulas for the Hilbert series of the Garsia-Haiman diagonal harmonics modules. These formulas involve weighted sums of labelled Dyck paths (or parking functions) relative to suitable statistics. This article introduces a third combinatorial formula that is shown to be equivalent to the first two. We show that the four statistics on labelled Dyck paths appearing in these formulas all have the same univariate distribution, which settles an earlier question of Haglund and Loehr. We then introduce analogous statistics on other collections of labelled lattice paths contained in trapezoids. We obtain a fermionic formula for the generating function for these statistics. We give bijective proofs of the equivalence of several forms of this generating function. These bijections imply that all the new statistics have the same univariate distribution. Using these new statistics, we conjecture combinatorial formulas for the Hilbert series of certain generalizations of the diagonal harmonics modules.

scholarly journals Standard fillings to parking functions

2012 ◽  
Vol DMTCS Proceedings vol. AR,... (Proceedings) ◽  
Author(s):  
Elizabeth Niese
Keyword(s):  
Generating Function ◽  
Hilbert Series ◽  
Parking Functions ◽  
Diagonal Harmonics ◽  
International Audience ◽  
Ferrers Diagrams

International audience The Hilbert series of the Garsia-Haiman module can be written as a generating function of standard fillings of Ferrers diagrams. It is conjectured by Haglund and Loehr that the Hilbert series of the diagonal harmonics can be written as a generating function of parking functions. In this paper we present a weight-preserving injection from standard fillings to parking functions for certain cases. La série Hilbert du module Garsia-Haiman peut être écrite comme fonction génératrice de tableaux des diagrammes Ferrers. Haglund et Loehr conjecturent que la série Hilbert de l'harmonic diagonale peut être écrite comme fonction génératrice des fonctions parking. Dans cet essai nous présentons une injection des tableaux vers les fonctions parking pour certains cas.

scholarly journals $h$-Polynomials via Reduced Forms

10.37236/5172 ◽  
2015 ◽  
Vol 22 (4) ◽  
Author(s):  
Karola Meszaros
Keyword(s):  
Reduced Form ◽  
Flow Polytopes ◽  
Flow Polytope ◽  
Reduced Forms

The flow polytope $F_{\widetilde{G}}$ is the set of nonnegative unit flows on the graph $\widetilde{G}$. The subdivision algebra of flow polytopes prescribes a way to dissect a flow polytope $F_{\widetilde{G}}$ into simplices. Such a dissection is encoded by the terms of the so called reduced form of the monomial $\prod_{(i,j)\in E(G)}x_{ij}$. We prove that we can use the subdivision algebra of flow polytopes to construct not only dissections, but also regular flag triangulations of flow polytopes. We prove that reduced forms in the subdivision algebra are generalizations of $h$-polynomials of the triangulations of flow polytopes. We deduce several corollaries of the above results, most notably proving certain cases of a conjecture of Kirillov about the nonnegativity of reduced forms in the noncommutative quasi-classical Yang-Baxter algebra.

scholarly journals Volumes of Flow Polytopes Related to Caracol Graphs

10.37236/9187 ◽  
2020 ◽  
Vol 27 (4) ◽  
Author(s):  
Jihyeug Jang ◽  
Jang Soo Kim
Keyword(s):  
Constant Term ◽  
Original Graph ◽  
Dyck Paths ◽  
Flow Vector ◽  
Flow Polytopes ◽  
Flow Polytope

Recently, Benedetti et al. introduced an Ehrhart-like polynomial associated to a graph. This polynomial is defined as the volume of a certain flow polytope related to a graph and has the property that the leading coefficient is the volume of the flow polytope of the original graph with net flow vector $(1,1,\dots,1)$. Benedetti et al. conjectured a formula for the Ehrhart-like polynomial of what they call a caracol graph. In this paper their conjecture is proved using constant term identities, labeled Dyck paths, and a cyclic lemma.

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