scholarly journals A Schröder Generalization of Haglund's Statistic on Catalan Paths

10.37236/1709 ◽  
2003 ◽  
Vol 10 (1) ◽  
Author(s):  
E. S. Egge ◽  
J. Haglund ◽  
K. Killpatrick ◽  
D. Kremer
Keyword(s):  
Symmetric Functions ◽  
Rational Functions ◽  
Lattice Paths ◽  
Nabla Operator ◽  
Special Values ◽  
Schröder Paths ◽  
Macdonald Symmetric Functions

Garsia and Haiman (J. Algebraic. Combin. $\bf5$ $(1996)$, $191-244$) conjectured that a certain sum $C_n(q,t)$ of rational functions in $q,t$ reduces to a polynomial in $q,t$ with nonnegative integral coefficients. Haglund later discovered (Adv. Math., in press), and with Garsia proved (Proc. Nat. Acad. Sci. ${\bf98}$ $(2001)$, $4313-4316$) the refined conjecture $C_n(q,t) = \sum q^{{\rm area}}t^{{\rm bounce}}$. Here the sum is over all Catalan lattice paths and ${\rm area}$ and ${\rm bounce}$ have simple descriptions in terms of the path. In this article we give an extension of $({\rm area},{\rm bounce})$ to Schröder lattice paths, and introduce polynomials defined by summing $q^{{\rm area}}t^{{\rm bounce}}$ over certain sets of Schröder paths. We derive recurrences and special values for these polynomials, and conjecture they are symmetric in $q,t$. We also describe a much stronger conjecture involving rational functions in $q,t$ and the $\nabla$ operator from the theory of Macdonald symmetric functions.

scholarly journals Limits of Modified Higher $q,t$-Catalan Numbers

10.37236/3201 ◽  
2013 ◽  
Vol 20 (3) ◽  
Author(s):  
Kyungyong Lee ◽  
Li Li ◽  
Nicholas A Loehr
Keyword(s):  
Representation Theory ◽  
Symmetric Functions ◽  
Rational Functions ◽  
Catalan Numbers ◽  
Integer Partitions ◽  
Hilbert Schemes ◽  
Intensive Study ◽  
Nabla Operator ◽  
Dyck Paths ◽  
Partition Statistics

The $q,t$-Catalan numbers can be defined using rational functions, geometry related to Hilbert schemes, symmetric functions, representation theory, Dyck paths, partition statistics, or Dyck words. After decades of intensive study, it was eventually proved that all these definitions are equivalent. In this paper, we study the similar situation for higher $q,t$-Catalan numbers, where the equivalence of the algebraic and combinatorial definitions is still conjectural. We compute the limits of several versions of the modified higher $q,t$-Catalan numbers and show that these limits equal the generating function for integer partitions. We also identify certain coefficients of the higher $q,t$-Catalan numbers as enumerating suitable integer partitions, and we make some conjectures on the homological significance of the Bergeron-Garsia nabla operator.

scholarly journals Cumulants of Jack symmetric functions and b-conjecture (extended abstract)

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Maciej Dolega ◽  
Valentin Féray
Keyword(s):  
Nonnegative Integer ◽  
Symmetric Functions ◽  
Rational Functions ◽  
Independent Interest ◽  
Jack Polynomials ◽  
Factorization Property ◽  
Integer Coefficients ◽  
Continuous Deformation ◽  
Generating Series ◽  
International Audience

International audience Goulden and Jackson (1996) introduced, using Jack symmetric functions, some multivariate generating series ψ(x, y, z; t, 1 + β) that might be interpreted as a continuous deformation of the rooted hypermap generating series. They made the following conjecture: coefficients of ψ(x, y, z; t, 1+β) are polynomials in β with nonnegative integer coefficients. We prove partially this conjecture, nowadays called b-conjecture, by showing that coefficients of ψ(x, y, z; t, 1 + β) are polynomials in β with rational coefficients. Until now, it was only known that they are rational functions of β. A key step of the proof is a strong factorization property of Jack polynomials when α → 0 that may be of independent interest.

scholarly journals Generating functions of planar polygons from homological mirror symmetry of elliptic curves

2020 ◽  
Vol 6 (3) ◽  
Author(s):  
Kathrin Bringmann ◽  
Jonas Kaszian ◽  
Jie Zhou
Keyword(s):  
Elliptic Curves ◽  
Theta Function ◽  
Generating Functions ◽  
Mirror Symmetry ◽  
Rational Functions ◽  
Mock Theta Function ◽  
Homological Mirror Symmetry ◽  
Jacobi Theta Function ◽  
Special Values

Abstract We study generating functions of certain shapes of planar polygons arising from homological mirror symmetry of elliptic curves. We express these generating functions in terms of rational functions of the Jacobi theta function and Zwegers’ mock theta function and determine their (mock) Jacobi properties. We also analyze their special values and singularities, which are of geometric interest as well.

scholarly journals JACK–LAURENT SYMMETRIC FUNCTIONS FOR SPECIAL VALUES OF PARAMETERS

2015 ◽  
Vol 58 (3) ◽  
pp. 599-616 ◽  
Author(s):  
A. N. SERGEEV ◽  
A. P. VESELOV
Keyword(s):  
Symmetric Functions ◽  
Natural Numbers ◽  
Linear Combinations ◽  
Special Values

AbstractWe consider the Jack–Laurent symmetric functions for special values of parametersp0=n+k−1m, wherekis not rational andmandnare natural numbers. In general, the coefficients of such functions may have poles at these values ofp0. The action of the corresponding algebra of quantum Calogero–Moser integrals$\mathcal{D}$(k,p0) on the space of Laurent symmetric functions defines the decomposition into generalised eigenspaces. We construct a basis in each generalised eigenspace as certain linear combinations of the Jack–Laurent symmetric functions, which are regular atp0=n+k−1m, and describe the action of$\mathcal{D}$(k,p0) in these eigenspaces.

scholarly journals Lax Operator for Macdonald Symmetric Functions

2015 ◽  
Vol 105 (7) ◽  
pp. 901-916 ◽  
Author(s):  
Maxim Nazarov ◽  
Evgeny Sklyanin
Keyword(s):  
Symmetric Functions ◽  
Lax Operator ◽  
Macdonald Symmetric Functions

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 Vertically Constrained Motzkin-Like Paths Inspired by Bobbin Lace

10.37236/7799 ◽  
2019 ◽  
Vol 26 (2) ◽  
Author(s):  
Veronika Irvine ◽  
Stephen Melczer ◽  
Frank Ruskey
Keyword(s):  
Mathematical Model ◽  
Generating Functions ◽  
Finite Lattice ◽  
Lattice Paths ◽  
Change Direction ◽  
Quarter Plane ◽  
Directed Paths ◽  
Schröder Paths ◽  
Motzkin Paths

Inspired by a new mathematical model for bobbin lace, this paper considers finite lattice paths formed from the set of step vectors $\mathfrak{A}=$$\{\rightarrow,$ $\nearrow,$ $\searrow,$ $\uparrow,$ $\downarrow\}$ with the restriction that vertical steps $(\uparrow, \downarrow)$ cannot be consecutive. The set $\mathfrak{A}$ is the union of the well known Motzkin step vectors $\mathfrak{M}=$$\{\rightarrow,$ $\nearrow,$ $\searrow\}$ with the vertical steps $\{\uparrow, \downarrow\}$. An explicit bijection $\phi$ between the exhaustive set of vertically constrained paths formed from $\mathfrak{A}$ and a bisection of the paths generated by $\mathfrak{M}S$ is presented. In a similar manner, paths with the step vectors $\mathfrak{B}=$$\{\nearrow,$ $\searrow,$ $\uparrow,$ $\downarrow\}$, the union of Dyck step vectors and constrained vertical steps, are examined.  We show, using the same $\phi$ mapping, that there is a bijection between vertically constrained $\mathfrak{B}$ paths and the subset of Motzkin paths avoiding horizontal steps at even indices.  Generating functions are derived to enumerate these vertically constrained, partially directed paths when restricted to the half and quarter-plane.  Finally, we extend Schröder and Delannoy step sets in a similar manner and find a bijection between these paths and a subset of Schröder paths that are smooth (do not change direction) at a regular horizontal interval.

scholarly journals Integrable probability: stochastic vertex models and symmetric functions

2018 ◽  
Author(s):  
Alexei Borodin ◽  
Leonid Petrov
Keyword(s):  
Symmetric Functions ◽  
Height Function ◽  
Rational Functions ◽  
Universality Class ◽  
Integral Representations ◽  
Baxter Equation ◽  
Partition Functions ◽  
Vertex Model ◽  
Schur Polynomials ◽  
Higher Spin

This chapter presents the study of a homogeneous stochastic higher spin six-vertex model in a quadrant. For this model concise integral representations for multipoint q-moments of the height function and for the q-correlation functions are derived. At least in the case of the step initial condition, these formulas degenerate in appropriate limits to many known formulas of such type for integrable probabilistic systems in the (1+1)d KPZ universality class, including the stochastic six-vertex model, ASEP, various q-TASEPs, and associated zero-range processes. The arguments are largely based on properties of a family of symmetric rational functions that can be defined as partition functions of the higher spin six-vertex model for suitable domains; they generalize classical Hall–Littlewood and Schur polynomials. A key role is played by Cauchy-like summation identities for these functions, which are obtained as a direct corollary of the Yang–Baxter equation for the higher spin six-vertex model.

scholarly journals REFINED BPS STATE COUNTING FROM NEKRASOV'S FORMULA AND MACDONALD FUNCTIONS

2009 ◽  
Vol 24 (12) ◽  
pp. 2253-2306 ◽  
Author(s):  
HIDETOSHI AWATA ◽  
HIROAKI KANNO
Keyword(s):  
Partition Function ◽  
Hilbert Scheme ◽  
Symmetric Functions ◽  
Topological Vertex ◽  
Yang Mills ◽  
Hilbert Scheme Of Points ◽  
Macdonald Symmetric Functions ◽  
M Theory ◽  
The Web ◽  
Refined Version

It has been argued that Nekrasov's partition function gives the generating function of refined BPS state counting in the compactification of M theory on local Calabi–Yau spaces. We show that a refined version of the topological vertex we previously proposed (arXiv:hep-th/0502061) is a building block of Nekrasov's partition function with two equivariant parameters. Compared with another refined topological vertex by Iqbal, Kozcaz and Vafa (arXiv:hep-th/0701156), our refined vertex is expressed entirely in terms of the specialization of the Macdonald symmetric functions which is related to the equivariant character of the Hilbert scheme of points on ℂ2. We provide diagrammatic rules for computing the partition function from the web diagrams appearing in geometric engineering of Yang–Mills theory with eight supercharges. Our refined vertex has a simple transformation law under the flop operation of the diagram, which suggests that homological invariants of the Hopf link are related to the Macdonald functions.

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