Jack characters and enumeration of maps

On Kerov polynomials for Jack characters

We consider a deformation of Kerov character polynomials, linked to Jack symmetric functions. It has been introduced recently by M. Lassalle, who formulated several conjectures on these objects, suggesting some underlying combinatorics. We give a partial result in this direction, showing that some quantities are polynomials in the Jack parameter $α$ with prescribed degree. Our result has several interesting consequences in various directions. Firstly, we give a new proof of the fact that the coefficients of Jack polynomials expanded in the monomial or power-sum basis depend polynomially in $α$…

An algorithm for map enumeration

Bauer and Itzykson showed that associated to each labeled map embedded on an oriented Riemann surface there was a group generated by a pair of permutations. From this result an algorithm may be constructed for enumerating labeled maps, and this construction is easily augmented to bin the numbers by the genus of the surface the map is embedded in. The results agree with the calculations of Harer and Zagier of 1-vertex maps; with those of Bessis, Itzykson, and Zuber of 4-valent maps; and with those of Ercolani, McLaughlin, and Pierce for $2ν$-valent maps. We then modify this algorithm to one whi…

Elliptic enumeration of nonintersecting lattice paths

We enumerate lattice paths in the planar integer lattice consisting of positively directed unit vertical and horizontal steps with respect to a specific elliptic weight function. The elliptic generating function of paths from a given starting point to a given end point evaluates to an elliptic generalization of the binomial coefficient. Convolution gives an identity equivalent to Frenkel and Turaev's 10-V-9 summation. This appears to be the first combinatorial proof of the latter, and at the same time of some important degenerate cases including Jackson's 8-phi-7 and Dougall's 7-F-6 summation.…

Jack polynomials and orientability generating series of maps

We study Jack characters, which are the coefficients of the power-sum expansion of Jack symmetric functions with a suitable normalization. These quantities have been introduced by Lassalle who formulated some challenging conjectures about them. We conjecture existence of a weight on non-oriented maps (i.e., graphs drawn on non-oriented surfaces) which allows to express any given Jack character as a weighted sum of some simple functions indexed by maps. We provide a candidate for this weight which gives a positive answer to our conjecture in some, but unfortunately not all, cases. In particular…

Polynomial properties of Jack connection coefficients and generalization of a result by Dénes

This article is devoted to the computation of Jack connection coefficients, a generalization of the connection coefficients of two classical commutative subalgebras of the group algebra of the symmetric group: the class algebra and the double coset algebra. The connection coefficients of these two algebraic structures are of significant interest in the study of Schur and zonal polynomials as well as the irreducible characters of the symmetric group and the zonal spherical functions. Furthermore they play an important role in combinatorics as they give the number of factorizations of a permutat…

利用Dumitriu-Edelman模型对地图的枚举

Enumeration of maps with the Dumitriu-Edelman model

利用Dumitriu-Edelman三对角模型，给出了β系综粒子幂和累积量在1/N和β下的展开，系数由Bouttier、Fusy和Guitter引入的适当标记地图表示，仅涉及可定向地图，并通过新映射将前两阶与LaCroix的展开联系起来。

Non-commutative Frobenius characteristic of generalized parking functions -- Application to enumeration

We give a recursive definition of generalized parking function that allows us to view them as a species. From there we compute a non-commutative characteristic of the generalized parking function module, and deduce some enumeration formulas of structures and isomorphism types. We give as well an interpretation in several bases of non-commutative symmetric functions. Finally, we investigate an inclusion-exclusion formula given by Kung and Yan.

Enumeration of Symmetry Classes of Alternating Sign Matrices and Characters of Classical Groups

An alternating sign matrix is a square matrix with entries 1, 0 and -1 such that the sum of the entries in each row and each column is equal to 1 and the nonzero entries alternate in sign along each row and each column. To some of the symmetry classes of alternating sign matrices and their variations, G. Kuperberg associate square ice models with appropriate boundary conditions, and give determinanat and Pfaffian formulae for the partition functions. In this paper, we utilize several determinant and Pfaffian identities to evaluate Kuperberg's determinants and Pfaffians, and express the round p…

Moments of characteristic polynomials for compact symmetric spaces and Jack polynomials

We express the averages of products of characteristic polynomials for random matrix ensembles associated with compact symmetric spaces in terms of Jack polynomials or Heckman and Opdam's Jacobi polynomials depending on the root system of the space. We also give explicit expressions for the asymptotic behavior of these averages in the limit as the matrix size goes to infinity.

The enumeration of maximally clustered permutations

The maximally clustered permutations are characterized by avoiding the classical permutation patterns 3421, 4312, and 4321. This class contains the freely-braided permutations and the fully-commutative permutations. In this work, we show that the generating functions for certain fully-commutative pattern classes can be transformed to give generating functions for the corresponding freely-braided and maximally clustered pattern classes. Moreover, this transformation of generating functions is rational. As a result, we obtain enumerative formulas for the pattern classes mentioned above as well a…

移位麦克唐纳多项式与(q,t)变形的古尔德恩-杰克逊乘积

Shifted Macdonald Polynomials and the $(q,t)$-Deformed Goulden--Jackson Product

本文研究移位麦克唐纳多项式与(q,t)变形古尔德恩-杰克逊乘积，构造实现移位麦克唐纳本征值的无穷级数，推导纳扎罗夫-斯克莱宁算子相关级数并与已有算子比较。

Gaussian fluctuations of Young diagrams and structure constants of Jack characters

In this paper, we consider a deformation of Plancherel measure linked to Jack polynomials. Our main result is the description of the first and second-order asymptotics of the bulk of a random Young diagram under this distribution, which extends celebrated results of Vershik-Kerov and Logan-Shepp (for the first order asymptotics) and Kerov (for the second order asymptotics). This gives more evidence of the connection with Gaussian $β$-ensemble, already suggested by some work of Matsumoto. Our main tool is a polynomiality result for the structure constant of some quantities that we call Jack cha…

Pieri Integral Formula and Asymptotics of Jack Unitary Characters

We introduce Jack (unitary) characters and prove two kinds of formulas that are suitable for their asymptotics, as the lengths of the signatures that parametrize them go to infinity. The first kind includes several integral representations for Jack characters of one variable. The second identity we prove is the Pieri integral formula for Jack characters which, in a sense, is dual to the well known Pieri rule for Jack polynomials. The Pieri integral formula can also be seen as a functional equation for irreducible spherical functions of virtual Gelfand pairs. As an application of our formulas, …

On the Enumeration of Skew Young Tableaux

We extend work of McKay, Morse, and Wilf by giving exact formulas and asymptotic formulas for the number of skew Young tableaux T in two situations: (1) the "inside shape" and total number of cells of T are fixed, and (2) the inside shape of T is fixed. The proofs use the theory of symmetric functions and estimates for irreducible characters of the symmetric group due to Thoma, Vershik, Kerov, and Biane.

加权象限游走的枚举：代数性与D-有限性的判据

Enumeration of weighted quadrant walks: criteria for algebraicity and D-finiteness

本文提出基于椭圆函数理论的统一方法，为象限内加权游走生成函数的代数性与D-有限性判定提供统一证明。

Enumeration of Tree-like Maps with Arbitrary Number of Vertices

This paper provides the generating series for the embedding of tree-like graphs of arbitrary number of vertices, accourding to their genus. It applies and extends the techniques of Chan, where it was used to give an alternate proof of the Goulden and Slofstra formula. Furthermore, this greatly generalizes the famous Harer-Zagier formula, which computes the Euler characteristic of the moduli space of curves, and is equivalent to the computation of one vertex maps.

Quasitrivial semigroups: characterizations and enumerations

We investigate the class of quasitrivial semigroups and provide various characterizations of the subclass of quasitrivial and commutative semigroups as well as the subclass of quasitrivial and order-preserving semigroups. We also determine explicitly the sizes of these classes when the semigroups are defined on finite sets. As a byproduct of these enumerations, we obtain several new integer sequences.

经典Weyl群中的高次Lie特征标与根计数

Higher Lie characters and root enumeration in classical Weyl groups

本研究证明D型经典Weyl群的k次根计数子为真特征标，核心方法是借助B型高次Lie特征标完成证明，推进了Weyl群表示论与根计数相关研究。

On Constructions and Enumeration of Vectorial Hyper-bent Functions in the $\cP\cS_{ap}^{\#}$ Class

The purpose of this paper is to give explicit constructions of vectorial hyper-bent functions in the $\cP\cS_{ap}^{\#}$ class. It seems that the explicit constructions were so far known only for very special cases. To this end, we present a sufficient and necessary condition of this family of vectorial functions to be hyper-bent. The conditions are expressed in terms of group ring. Using this characterization, explicit constructions of vectorial hyper-bent functions of the $\cP\cS_{ap}^{\#}$ class via balanced functions are proposed. Furthermore, exact number of vectorial hyper-bent functions …

A Generalization of Polya's Enumeration Theorem or the Secret Life of Certain Index Sets

Polya's fundamental enumeration theorem is generalized in terms of Schur-Macdonald's theory (S-MT) of invariant matrices. Given a permutation group $W\leq S_d$ and a one-dimensional character $χ$ of $W$, the polynomial functor $F_χ$ corresponding via S-MT to the induced monomial representation $U_χ= ind_W^{S_d}(χ)$ of $S_d$, is studied. It turns out that the characteristic $ch(F_χ)$ is the weighted inventory of some set $J(χ)$ of $W$-orbits in the integer-valued hypercube $[0,\infty)^d$. The elements of $J(χ) can be distinguished among all $W$-orbits by a maximum property. The identity $ch(F_χ…

关于置换不变与完备那不勒斯停车函数的枚举

On the enumeration of permutation-invariant and complete Naples parking functions

本文提出基于自然组合分解的有效枚举方法，用于计数置换不变与完备那不勒斯停车函数，得到递归可描述的公式。

Different classes of binary necklaces and a combinatorial method for their enumerations

In this paper we investigate enumeration of some classes of $n$-character strings and binary necklaces. Recall that binary necklaces are necklaces in two colors with length $n$. We prove three results (Theorems 1, 1' and 2) concerning the numbers of three classes of $j$-character strings (closely related to some classes of binary necklaces or Lyndon words). Using these results, we deduce Moreau's necklace-counting function for binary aperiodic necklaces of length $k$ \cite{mo} (Theorem 3), and we prove the binary case of MacMahon's formula from 1892 \cite{ma} (also called Witt's formula) for t…

Enumeration of parallelogram polycubes

In this paper, we enumerate parallelogram polycubes according to several parameters. After establishing a relation between Multiple Zeta Function and the Dirichlet generating function of parallelogram polyominoes, we generalize it to the case of parallelogram polycubes. We also give an explicit formula and an ordinary generating function of parallelogram polycubes according to the width, length and depth, by characterizing its projections. Then, these results are generalized to polyhypercubes.

Enumeration of Commuting Pairs in Lie Algebras over Finite Fields

Feit and Fine derived a generating function for the number of ordered pairs of commuting n by n matrices over the finite field F_q. This has been reproved and studied by Bryan and Morrison from the viewpoint of motivic Donaldson-Thomas theory. In this note we give a new proof of the Feit-Fine result, and generalize it to the Lie algebra of finite unitary groups and to the Lie algebra of odd characteristic finite symplectic groups. We extract some asymptotic information from these generating functions. Finally, we derive generating functions for the number of commuting nilpotent elements for th…

素数$p$下自同构群为$\PGam(2,2^p)$的可定向正则地图的枚举

Enumeration of orientably-regular maps with automorphism group $\PGam(2,2^p)$ for prime $p$

利用群特征标，枚举了自同构群为$\PGam(2,2^p)$（$p$为素数）的给定类型可定向正则地图。

Combinatorial methods of character enumeration for the unitriangular group

Let $\UT_n(q)$ denote the group of unipotent $n\times n$ upper triangular matrices over a field with $q$ elements. The degrees of the complex irreducible characters of $\UT_n(q)$ are precisely the integers $q^e$ with $0\leq e\leq \lfloor \frac{n}{2} \rfloor \lfloor \frac{n-1}{2} \rfloor$, and it has been conjectured that the number of irreducible characters of $\UT_n(q)$ with degree $q^e$ is a polynomial in $q-1$ with nonnegative integer coefficients (depending on $n$ and $e$). We confirm this conjecture when $e\leq 8$ and $n$ is arbitrary by a computer calculation. In particular, we describe …

有根二叉完美系统发育树的计数

Enumeration of rooted binary perfect phylogenies

本文研究了有根二叉完美系统发育树的计数问题，通过将大小为s的谱系样本分布到n个叶子的无标签树上，推广了有根二叉无标签树并刻画了其组合结构。

仅偶数部分可加上划线的overpartition模4枚举

Enumeration modulo 4 of overpartitions wherein only even parts can be overlined

本文为Munagi和Sellers关于仅偶数部分可加上划线的overpartition数量 $d_2(n)$ 的奇偶性特征提供了组合证明，并将其扩展至模4特征，结合经典对合与全新方法给出了生成函数和组合双重证明。

Polya枚举，缠绕积对称函数，以及曲线的模空间

Pólya enumeration, wreath product symmetric functions, and moduli spaces of curves

本文提出了一种基于缠绕积对称函数的框架，用于计算稳定曲线模空间的$S_n$-等价欧拉特征，并通过Adams操作和幂和扭曲建立其与普通对称函数的关系。

曲线与映射模空间的图枚举

Graph enumeration for moduli spaces of curves and maps

本文提出基于图枚举的微积分，用于计算图形分层模空间的等变动机不变量，并应用于Deligne--Mumford模空间及稳定映射空间，借助Pólya--Petersen特征和花环积对称函数代数推导新公式。

Random Matrices, Graphical Enumeration and the Continuum Limit of Toda Lattices

In this paper we derive a hierarchy of differential equations which uniquely determine the coefficients in the asymptotic expansion, for large $N$, of the logarithm of the partition function of $N \times N$ Hermitian random matrices. These coefficients are generating functions for graphical enumeration on Riemann surfaces. The case that we particularly consider is for an underlying measure that differs from the Gaussian weight by a single monomial term of degree $2ν$. The coupling parameter for this term plays the role of the independent dynamical variable in the differential equations. From t…

欧拉地图的通用渐近性与精确计数

Universal Asymptotics and Exact Enumeration of Eulerian Maps

研究具有任意度序列的连通、带标记、亏格为 \(g\) 的欧拉地图数量渐近性，结合正交多项式递推系数分析与多变量解析组合学理论，得出通用渐近性及亏格为 \(1\) 时的精确公式，是 \(g\geq 1\) 非正则地图计数问题的首个结果。