scholarly journals The Delta Conjecture

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
James Haglund ◽  
Jeffrey B. Remmel ◽  
Andrew Timothy Wilson
Keyword(s):  
Symmetric Function ◽  
Ordered Set ◽  
Macdonald Polynomials ◽  
Catalan Numbers ◽  
Set Partitions ◽  
Combinatorial Interpretations ◽  
Llt Polynomials ◽  
Ordered Set Partitions ◽  
International Audience ◽  
Tesler Matrices

International audience We conjecture two combinatorial interpretations for the symmetric function ∆eken, where ∆f is an eigenoperator for the modified Macdonald polynomials defined by Bergeron, Garsia, Haiman, and Tesler. Both interpretations can be seen as generalizations of the Shuffle Conjecture, a statement originally conjectured by Haglund, Haiman, Remmel, Loehr, and Ulyanov and recently proved by Carlsson and Mellit. We show how previous work of the second and third authors on Tesler matrices and ordered set partitions can be used to verify several cases of our conjectures. Furthermore, we use a reciprocity identity and LLT polynomials to prove another case. Finally, we show how our conjectures inspire 4-variable generalizations of the Catalan numbers, extending work of Garsia, Haiman, and the first author.

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.

Euler–Mahonian Statistics on Ordered Set Partitions

2008 ◽  
Vol 22 (3) ◽  
pp. 1105-1137 ◽  
Author(s):  
Masao Ishikawa ◽  
Anisse Kasraoui ◽  
Jiang Zeng
Keyword(s):  
Ordered Set ◽  
Set Partitions ◽  
Ordered Set Partitions ◽  
Mahonian Statistics

scholarly journals The polytope of Tesler matrices

2015 ◽  
Vol DMTCS Proceedings, 27th... (Proceedings) ◽  
Author(s):  
Karola Mészáros ◽  
Alejandro H. Morales ◽  
Brendon Rhoades
Keyword(s):  
Partition Function ◽  
Catalan Numbers ◽  
Young Tableaux ◽  
Integer Points ◽  
International Audience ◽  
Kostant Partition Function ◽  
Tesler Matrices ◽  
Flow Polytope ◽  
Standard Young Tableaux

26 pages, 4 figures. v2 has typos fixed, updated references, and a final remarks section including remarks from previous sections International audience We introduce the Tesler polytope $Tes_n(a)$, whose integer points are the Tesler matrices of size n with hook sums $a_1,a_2,...,a_n in Z_{\geq 0}$. We show that $Tes_n(a)$ is a flow polytope and therefore the number of Tesler matrices is counted by the type $A_n$ Kostant partition function evaluated at $(a_1,a_2,...,a_n,-\sum_{i=1}^n a_i)$. We describe the faces of this polytope in terms of "Tesler tableaux" and characterize when the polytope is simple. We prove that the h-vector of $Tes_n(a)$ when all $a_i>0$ is given by the Mahonian numbers and calculate the volume of $Tes_n(1,1,...,1)$ to be a product of consecutive Catalan numbers multiplied by the number of standard Young tableaux of staircase shape. On présente le polytope de Tesler $Tes_n(a)$, dont les points réticuilaires sont les matrices de Tesler de taillen avec des sommes des équerres $a_1,a_2,...,a_n in Z_{\geq 0}$. On montre que $Tes_n(a)$ est un polytope de flux. Donc lenombre de matrices de Tesler est donné par la fonction de Kostant de type An évaluée à ($(a_1,a_2,...,a_n,-\sum_{i=1}^n a_i)$On décrit les faces de ce polytope en termes de “tableaux de Tesler” et on caractérise quand le polytope est simple.On montre que l’h-vecteur de $Tes_n(a)$ , quand tous les $a_i>0$ , est donnée par le nombre de permutations avec unnombre donné d’inversions et on calcule le volume de T$Tes_n(1,1,...,1)$ comme un produit de nombres de Catalanconsécutives multiplié par le nombre de tableaux standard de Young en forme d’escalier

scholarly journals An extension of MacMahon's equidistribution theorem to ordered set partitions

2015 ◽  
Vol 134 ◽  
pp. 242-277 ◽  
Author(s):  
Jeffrey B. Remmel ◽  
Andrew Timothy Wilson
Keyword(s):  
Ordered Set ◽  
Set Partitions ◽  
Ordered Set Partitions

scholarly journals Ordered set partitions and the $0$-Hecke algebra

2018 ◽  
Vol 1 (1) ◽  
pp. 47-80 ◽  
Author(s):  
Jia Huang ◽  
Brendon Rhoades
Keyword(s):  
Ordered Set ◽  
Hecke Algebra ◽  
Set Partitions ◽  
Ordered Set Partitions

scholarly journals Coloring Rings in Species

2014 ◽  
Vol DMTCS Proceedings vol. AT,... (Proceedings) ◽  
Author(s):  
Jacob White
Keyword(s):  
Directed Graph ◽  
Symmetric Function ◽  
Chromatic Polynomial ◽  
Connected Components ◽  
Expansion Formula ◽  
Strongly Connected Components ◽  
Set Partitions ◽  
Strongly Connected ◽  
International Audience ◽  
Chromatic Symmetric Function

International audience We present a generalization of the chromatic polynomial, and chromatic symmetric function, arising in the study of combinatorial species. These invariants are defined for modules over lattice rings in species. The primary examples are graphs and set partitions. For these new invariants, we present analogues of results regarding stable partitions, the bond lattice, the deletion-contraction recurrence, and the subset expansion formula. We also present two detailed examples, one related to enumerating subgraphs by their blocks, and a second example related to enumerating subgraphs of a directed graph by their strongly connected components.

scholarly journals Hall-Littlewood polynomials and a Hecke action on ordered set partitions

2019 ◽  
Vol 147 (5) ◽  
pp. 1839-1850
Author(s):  
Jia Huang ◽  
Brendon Rhoades ◽  
Travis Scrimshaw
Keyword(s):  
Ordered Set ◽  
Set Partitions ◽  
Littlewood Polynomials ◽  
Ordered Set Partitions

scholarly journals Pattern Avoidance in Ordered Set Partitions

2014 ◽  
Vol 18 (3) ◽  
pp. 429-445 ◽  
Author(s):  
Anant Godbole ◽  
Adam Goyt ◽  
Jennifer Herdan ◽  
Lara Pudwell
Keyword(s):  
Ordered Set ◽  
Pattern Avoidance ◽  
Set Partitions ◽  
Ordered Set Partitions

scholarly journals Euler–Mahonian statistics on ordered set partitions (II)

2009 ◽  
Vol 116 (3) ◽  
pp. 539-563 ◽  
Author(s):  
Anisse Kasraoui ◽  
Jiang Zeng
Keyword(s):  
Ordered Set ◽  
Set Partitions ◽  
Ordered Set Partitions ◽  
Mahonian Statistics
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