scholarly journals Top Degree Part in $b$-Conjecture for Unicellular Bipartite Maps

10.37236/6130 ◽  
2017 ◽  
Vol 24 (3) ◽  
Author(s):  
Maciej Dołęga
Keyword(s):  
Positive Integer ◽  
Symmetric Functions ◽  
White Vertex ◽  
Additional Parameter ◽  
Combinatorial Interpretation ◽  
Integer Coefficients ◽  
Special Values ◽  
Generating Series ◽  
Special Value ◽  
Vertex Degrees

Goulden and Jackson (1996) introduced, using Jack symmetric functions, some multivariate generating series $\psi(\boldsymbol{x}, \boldsymbol{y},\boldsymbol{z}; 1, 1+\beta)$ with an additional parameter $\beta$ that might be interpreted as a continuous deformation of the rooted bipartite maps generating series. Indeed, it has a property that for $\beta \in \{0,1\}$, it specializes to the rooted, orientable (general, i.e. orientable or not, respectively) bipartite maps generating series. They made the following conjecture: coefficients of $\psi$ are polynomials in $\beta$ with positive integer coefficients that can be written as a multivariate generating series of rooted, general bipartite maps, where the exponent of $\beta$ is an integer-valued statistics that in some sense "measures the non-orientability" of the corresponding bipartite map.We show that except two special values of $\beta = 0,1$ for which the combinatorial interpretation of the coefficients of $\psi$ is known, there exists a third special value $\beta = -1$ for which the coefficients of $\psi$ indexed by two partitions $\mu,\nu$, and one partition with only one part are given by rooted, orientable bipartite maps with arbitrary face degrees and black/white vertex degrees given by $\mu$/$\nu$, respectively. We show that this evaluation corresponds, up to a sign, to a top-degree part of the coefficients of $\psi$. As a consequence, we introduce a collection of integer-valued statistics of maps $(\eta)$ such that the top-degree of the multivariate generating series of rooted, bipartite maps with only one face (called unicellular) with respect to $\eta$ gives the top degree of the appropriate coefficients of $\psi$. Finally, we show that $b$ conjecture holds true for all rooted, unicellular bipartite maps of genus at most $2$.

scholarly journals SPECIAL VALUES OF DIRICHLET SERIES AND ZETA INTEGRALS

2012 ◽  
Vol 08 (03) ◽  
pp. 697-714 ◽  
Author(s):  
EDUARDO FRIEDMAN ◽  
ALDO PEREIRA
Keyword(s):  
Positive Integer ◽  
Analytic Continuation ◽  
Dirichlet Series ◽  
Direct Calculation ◽  
Simple Relation ◽  
Product Rule ◽  
Special Values ◽  
Special Value

For f and g polynomials in p variables, we relate the special value at a non-positive integer s = -N, obtained by analytic continuation of the Dirichlet series [Formula: see text], to special values of zeta integrals Z(s;f,g) = ∫x∊[0, ∞)p g(x)f(x)-s dx ( Re (s) ≫ 0). We prove a simple relation between ζ(-N;f,g) and Z(-N;fa, ga), where for a ∈ ℂp, fa(x) is the shifted polynomial fa(x) = f(a + x). By direct calculation we prove the product rule for zeta integrals at s = 0, degree (fh) ⋅ Z(0;fh, g) = degree (f) ⋅ Z(0;f, g) + degree (h) ⋅ Z(0;h, g), and deduce the corresponding rule for Dirichlet series at s = 0, degree (fh) ⋅ ζ(0;fh, g) = degree (f) ⋅ ζ(0;f, g)+ degree (h)⋅ζ(0;h, g). This last formula generalizes work of Shintani and Chen–Eie.

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 A symmetric Bloch–Okounkov theorem

2021 ◽  
Vol 8 (2) ◽  
Author(s):  
Jan-Willem M. van Ittersum
Keyword(s):  
Symmetric Functions ◽  
Graded Algebra ◽  
Symmetric Polynomials ◽  
Generating Series

AbstractThe algebra of so-called shifted symmetric functions on partitions has the property that for all elements a certain generating series, called the q-bracket, is a quasimodular form. More generally, if a graded algebra A of functions on partitions has the property that the q-bracket of every element is a quasimodular form of the same weight, we call A a quasimodular algebra. We introduce a new quasimodular algebra $$\mathcal {T}$$ T consisting of symmetric polynomials in the part sizes and multiplicities.

scholarly journals Extension of Strongly Regular Graphs

10.37236/878 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Ralucca Gera ◽  
Jian Shen
Keyword(s):  
Positive Integer ◽  
Nonnegative Integer ◽  
Regularity Property ◽  
Strongly Regular Graphs ◽  
Regular Graphs ◽  
Strongly Regular ◽  
Vertex Degrees

The Friendship Theorem states that if any two people in a party have exactly one common friend, then there exists a politician who is a friend of everybody. In this paper, we generalize the Friendship Theorem. Let $\lambda$ be any nonnegative integer and $\mu$ be any positive integer. Suppose each pair of friends have exactly $\lambda$ common friends and each pair of strangers have exactly $\mu$ common friends in a party. The corresponding graph is a generalization of strongly regular graphs obtained by relaxing the regularity property on vertex degrees. We prove that either everyone has exactly the same number of friends or there exists a politician who is a friend of everybody. As an immediate consequence, this implies a recent conjecture by Limaye et. al.

scholarly journals On the Error Term in Duke's Estimate for the Average Special Value of L-Functions

2005 ◽  
Vol 48 (4) ◽  
pp. 535-546 ◽  
Author(s):  
Jordan S. Ellenberg
Keyword(s):  
Error Term ◽  
Orthonormal Basis ◽  
Cusp Forms ◽  
Nonzero Constant ◽  
Weighted Averages ◽  
Special Values ◽  
Special Value

AbstractLet be an orthonormal basis for weight 2 cusp forms of level N. We show that various weighted averages of special values L( f ⭙ χ, 1) over f ∈ are equal to 4πc +O(N–1+∊), where c is an explicit nonzero constant. A previous result of Duke gives an error term of O(N–1/2 log N).

scholarly journals Statistics on Lattice Walks and q-Lassalle Numbers

2015 ◽  
Vol DMTCS Proceedings, 27th... (Proceedings) ◽  
Author(s):  
Lenny Tevlin
Keyword(s):  
Positive Integer ◽  
Generating Function ◽  
Catalan Numbers ◽  
Binomial Coefficients ◽  
Integer Coefficients ◽  
Major Index ◽  
Lattice Walks ◽  
International Audience ◽  
The Difference ◽  
Positive Integers

International audience This paper contains two results. First, I propose a $q$-generalization of a certain sequence of positive integers, related to Catalan numbers, introduced by Zeilberger, see Lassalle (2010). These $q$-integers are palindromic polynomials in $q$ with positive integer coefficients. The positivity depends on the positivity of a certain difference of products of $q$-binomial coefficients.To this end, I introduce a new inversion/major statistics on lattice walks. The difference in $q$-binomial coefficients is then seen as a generating function of weighted walks that remain in the upper half-plan. Cet document contient deux résultats. Tout d’abord, je vous propose un $q$-generalization d’une certaine séquence de nombres entiers positifs, liés à nombres de Catalan, introduites par Zeilberger (Lassalle, 2010). Ces $q$-integers sont des polynômes palindromiques à $q$ à coefficients entiers positifs. La positivité dépend de la positivité d’une certaine différence de produits de $q$-coefficients binomial.Pour ce faire, je vous présente une nouvelle inversion/major index sur les chemins du réseau. La différence de $q$-binomial coefficients est alors considérée comme une fonction de génération de trajets pondérés qui restent dans le demi-plan supérieur.

Inequalities for Generalized Symmetric Functions

1969 ◽  
Vol 12 (5) ◽  
pp. 615-623 ◽  
Author(s):  
K.V. Menon
Keyword(s):  
Symmetric Function ◽  
Symmetric Functions ◽  
Elementary Symmetric Function ◽  
Generating Series ◽  
Image Position ◽  
Complete Symmetric Function

The generating series for the elementary symmetric function Er, the complete symmetric function Hr, are defined byrespectively.

scholarly journals Formal Group Laws and Chromatic Symmetric Functions of Hypergraphs

2015 ◽  
Vol DMTCS Proceedings, 27th... (Proceedings) ◽  
Author(s):  
Jair Taylor
Keyword(s):  
Power Series ◽  
Symmetric Functions ◽  
Formal Group ◽  
Combinatorial Interpretation ◽  
Recursive Structure ◽  
Combinatorial Objects ◽  
Formal Group Law ◽  
International Audience ◽  
Formal Group Laws ◽  
Group Law

International audience If $f(x)$ is an invertible power series we may form the symmetric function $f(f^{-1}(x_1)+f^{-1}(x_2)+...)$ which is called a formal group law. We give a number of examples of power series $f(x)$ that are ordinary generating functions for combinatorial objects with a recursive structure, each of which is associated with a certain hypergraph. In each case, we show that the corresponding formal group law is the sum of the chromatic symmetric functions of these hypergraphs by finding a combinatorial interpretation for $f^{-1}(x)$. We conjecture that the chromatic symmetric functions arising in this way are Schur-positive. Si $f(x)$ est une série entière inversible, nous pouvons former la fonction symétrique $f(f^{-1}(x_1)+f^{-1}(x_2)+...)$ que nous appelons une loi de groupe formel. Nous donnons plusieurs exemples de séries entières $f(x)$ qui sont séries génératrices ordinaires pour des objets combinatoires avec une structure récursive, chacune desquelles est associée à un certain hypergraphe. Dans chaque cas, nous donnons une interprétation combinatoire à $f^{-1}(x)$, ce qui nous permet de montrer que la loi de groupe formel correspondante est la somme des fonctions symétriques chromatiques de ces hypergraphes. Nous conjecturons que les fonctions symétriques chromatiques apparaissant de cette manière sont Schur-positives.

scholarly journals Some Natural Bigraded $S_n$-Modules

10.37236/1282 ◽  
1996 ◽  
Vol 3 (2) ◽  
Author(s):  
A. M. Garsia ◽  
M. Haiman
Keyword(s):  
Positive Integer ◽  
Linear Span ◽  
Basic Properties ◽  
Integer Coefficients ◽  
Derivatives Of

We construct for each $\mu\vdash n $ a bigraded $S_n$-module $\mathbf{H}_\mu$ and conjecture that its Frobenius characteristic $C_{\mu}(x;q,t)$ yields the Macdonald coefficients $K_{\lambda\mu}(q,t)$. To be precise, we conjecture that the expansion of $C_{\mu}(x;q,t)$ in terms of the Schur basis yields coefficients $C_{\lambda\mu}(q,t)$ which are related to the $K_{\lambda\mu}(q,t)$ by the identity $C_{\lambda\mu}(q,t)=K_{\lambda\mu}(q,1/t)t^{n(\mu )}$. The validity of this would give a representation theoretical setting for the Macdonald basis $\{ P_\mu(x;q,t)\}_\mu$ and establish the Macdonald conjecture that the $K_{\lambda\mu}(q,t)$ are polynomials with positive integer coefficients. The space $\mathbf{H}_\mu$ is defined as the linear span of derivatives of a certain bihomogeneous polynomial $\Delta_\mu(x,y)$ in the variables $x_1,x_2,\ldots ,x_n$, $y_1,y_2,\ldots ,y_n$. On the validity of our conjecture $\mathbf{H}_\mu$ would necessarily have $n!$ dimension. We refer to the latter assertion as the $n!$-conjecture. Several equivalent forms of this conjecture will be discussed here together with some of their consequences. In particular, we derive that the polynomials $C_{\lambda\mu}(q,t)$ have a number of basic properties in common with the coefficients $\tilde{K}_{\lambda\mu}(q,t)=K_{\lambda\mu}(q,1/t)t^{n(\mu )}$. For instance, we show that $C_{\lambda\mu}(0,t)=\tilde{K}_{\lambda\mu}(0,t)$, $C_{\lambda\mu}(q,0)=\tilde{K}_{\lambda\mu}(q,0)$ and show that on the $n!$ conjecture we must also have the equalities $C_{\lambda\mu}(1,t)=\tilde{K}_{\lambda\mu}(1,t)$ and $C_{\lambda\mu}(q,1)=\tilde{K}_{\lambda\mu}(q,1)$. The conjectured equality $C_{\lambda\mu}(q,t)=K_{\lambda\mu}(q,1/t)t^{n(\mu )}$ will be shown here to hold true when $\lambda$ or $\mu$ is a hook. It has also been shown (see [9]) when $\mu$ is a $2$-row or $2$-column partition and in [18] when $\mu$ is an augmented hook.

scholarly journals On the Matchings-Jack Conjecture for Jack Connection Coefficients Indexed by Two Single Part Partitions

10.37236/5085 ◽  
2016 ◽  
Vol 23 (1) ◽  
Author(s):  
Andrei L. Kanunnikov ◽  
Ekaterina A. Vassilieva
Keyword(s):  
Arbitrary Number ◽  
Symmetric Functions ◽  
Recurrence Formula ◽  
Integer Partitions ◽  
Integer Coefficients ◽  
Power Sum ◽  
Connection Coefficients ◽  
Algebraic Framework ◽  
Commutative Subalgebras ◽  
Single Part

This article is devoted to the study of Jack connection coefficients, a generalization of the connection coefficients of the classical commutative subalgebras of the group algebra of the symmetric group closely related to the theory of Jack symmetric functions. First introduced by Goulden and Jackson (1996) these numbers indexed by three partitions of a given integer $n$ and the Jack parameter $\alpha$ are defined as the coefficients in the power sum expansion of some Cauchy sum for Jack symmetric functions. Goulden and Jackson conjectured that they are polynomials in $\beta = \alpha-1$ with non negative integer coefficients of combinatorial significance, the Matchings-Jack conjecture.In this paper we look at the case when two of the integer partitions are equal to the single part $(n)$. We use an algebraic framework of Lasalle (2008) for Jack symmetric functions and a bijective construction in order to show that the coefficients satisfy a simple recurrence formula and prove the Matchings-Jack conjecture in this case. Furthermore we exhibit the polynomial properties of more general coefficients where the two single part partitions are replaced by an arbitrary number of integer partitions either equal to $(n)$ or $[1^{n-2}2]$.

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