Beáta Bényi, Toshiki Matsusaka
In this paper we introduce three combinatorial models for symmetrized poly-Bernoulli numbers. Based on our models we derive generalizations of some identities for poly-Bernoulli numbers. Finally, we set open questions and directions of further studies.
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Poly-Bernoulli numbers are one of generalizations of the classical Bernoulli numbers. Since a negative index poly-Bernoulli number is an integer, it is an interesting problem to study this number from combinatorial viewpoint. In this short article, we give a new combinatorial relation between symmetrized poly-Bernoulli numbers and the Dumont-Foata polynomials.
Recently, Bényi and the second author introduced two combinatorial interpretations for symmetrized poly-Bernoulli polynomials. In the present study, we construct bijections between these combinatorial objects. We also define various combinatorial polynomials and prove that all of these polynomials coincide with symmetrized poly-Bernoulli polynomials.
We introduce poly-Cauchy permutations that are enumerated by the poly-Cauchy numbers. We provide combinatorial proofs for several identities involving poly-Cauchy numbers and some of their generalizations. The aim of this work is to demonstrate the power and beauty of the elementary combinatorial approach.
Arithmetic combinatorics is often concerned with the problem of bounding the behaviour of arbitrary finite sets in a group or ring with respect to arithmetic operations such as addition or multiplication. Similarly, combinatorial geometry is often concerned with the problem of bounding the behaviour of arbitrary finite collections of geometric objects such as points, lines, or circles with respect to geometric operations such as incidence or distance. Given the presence of arbitrary finite sets in these problems, the methods used to attack these problems have primarily been combinatorial in na…
A new shift operator-based polynomial method in additive combinatorics
本文提出基于移位算子的新多项式方法，统一并简化了Alon组合零点定理与Croot-Lev-Pach方法等结果，并展望了未来应用方向。
A combinatorial theory for type $R_I$ orthogonal polynomials is given. The ingredients include weighted generalized Motzkin paths, moments, continued fractions, determinants, and histories. Several explicit examples in the Askey scheme are given.
We introduce a family of tableaux that simultaneously generalizes the tableaux used to characterize Grothendieck polynomials and k-Schur functions. We prove that the polynomials drawn from these tableaux are the affine Grothendieck polynomials and k-K-Schur functions -- Schubert representatives for the K-theory of affine Grassmannians and their dual in the nil Hecke ring. We prove a number of combinatorial properties including Pieri rules.
Our main aim with these notes is to introduce the combinatorial and symmetric function tools that relate to the description of the Poincare polynomial of the triply graded Khovanov-Rozansky homology of torus links, a.k.a. the (reduced) superpolynomial of these links.
The ${\mathbb B}_n^{(k)}$ poly-Bernoulli numbers --- a natural generalization of classical Bernoulli numbers ($B_n={\mathbb B}_n^{(1)}$) --- were introduced by Kaneko in 1997. When the parameter $k$ is negative then ${\mathbb B}_n^{(k)}$ is a nonnegative number. Brewbaker was the first to give combinatorial interpretation of these numbers. He proved that ${\mathbb B}_n^{(-k)}$ counts the so called lonesum $0\text{-}1$ matrices of size $n\times k$. Several other interpretations were pointed out. We survey these and give new ones. Our new interpretation, for example, gives a transparent, combina…
Combinatorics behind discriminants of polynomial systems
本文提出了一种基于多面体的组合结构BK集，用于研究多项式系统判别式的组合学基础，并推广了经典判别式理论。
We describe various aspects of the Al-Salam-Chihara $q$-Laguerre polynomials. These include combinatorial descriptions of the polynomials, the moments, the orthogonality relation and a combinatorial interpretation of the linearization coefficients. It is remarkable that the corresponding moment sequence appears also in the recent work of Postnikov and Williams on enumeration of totally positive Grassmann cells.
We develop a combinatorial model of the associated Hermite polynomials and their moments, and prove their orthogonality with a sign-reversing involution. We find combinatorial interpretations of the moments as complete matchings, connected complete matchings, oscillating tableaux, and rooted maps and show weight-preserving bijections between these objects. Several identities, linearization formulas, the moment generating function, and a second combinatorial model are also derived.
Combinatorics of orthogonal polynomials on the unit circle
本文从组合视角研究单位圆上正交多项式的矩，给出三种路径解释，并据此推导圆Jacobi与Rogers–Szegő多项式的广义矩显式公式，还引入并解释了广义线性化系数。
We investigate coordination numbers of the cubic lattices with emphases on their analytic behaviors, including the total positivity of the coordination matrices, the distribution of zeros of the coordination polynomials, the asymptotic normality of the coefficients of the coordination polynomials, the log-concavity and the log-convexity of the coordination numbers.
We study some combinatorial and algebraic properties of certain quadratic algebras related with dynamical classical and classical Yang-Baxter equations. One can find more details about the content of present paper in Extended Abstract.
We show that the geometric lifting of the RSK correspondence introduced by A.N. Kirillov (2001) is volume preserving with respect to a natural product measure on its domain, and that the integrand in Givental's integral formula for GL(n,R)-Whittaker functions arises naturally in this context. Apart from providing further evidence that Whittaker functions are the natural analogue of Schur polynomials in this setting, our results also provide a new `combinatorial' framework for the study of random polymers. When the input matrix consists of random inverse gamma distributed weights, the probabili…
On Combinatorics of the Arthur Trace Formula, Convex Polytopes, and Toric Varieties
本文从扇与多面体出发构造组合截断函数，证明其积分的收敛性与多项式性，推广了Arthur迹公式、Ehrhart多项式及Langlands引理，并给出环面簇上的几何解释。
Let $ρ$ be a metric on the set $X=\{1,2,\dots,n+1\}$. Consider the $n$-dimensional polytope of functions $f:X\rightarrow \mathbb{R}$, which satisfy the conditions $f(n+1)=0$, $|f(x)-f(y)|\leq ρ(x,y)$. The question on classifying metrics depending on the combinatorics of this polytope have been recently posed by A. M. Vershik \cite{V}. We prove that for any "generic" metric the number of $(n-m)$-dimensional faces, $0\leq m\leq n$, equals $\binom{n+m}{m,m,n-m}=(n+m)!/m!m!(n-m)!$. This fact is intimately related to regular triangulations of the root polytope (the convex hull of the roots of $A_n$…
We propose a new approach to the combinatorial interpretations of linearization coefficient problem of orthogonal polynomials. We first establish a difference system and then solve it combinatorially and analytically using the method of separation of variables. We illustrate our approach by applying it to determine the number of perfect matchings, derangements, and other weighted permutation problems. The separation of variables technique naturally leads to integral representations of combinatorial numbers where the integrand contains a product of one or more types of orthogonal polynomials. T…
Blurred combinatorics in resolution of singularities: (a little) beyond the characteristic polytope
本文为代数超曲面引入 Newton-Hironaka 多面体的扰动变体，由实数参数化，在参数值合适时接近原多面体，并提供不依赖参数选择的额外信息。
In this paper we study the topology of the strata, indexed by number partitions $λ$, in the natural stratification of the space of monic hyperbolic polynomials of degree $n$. We prove stabilization theorems for removing an independent block or an independent relation in $λ$. We also prove contractibility of the one-point compactifications of the strata indexed by a large class of number partitions, including $λ=(k^m,1^r)$, for $m\geq 2$. Furthermore, we study the maps between the homology groups of the strata, induced by imposing additional relations (resonances) on the number partition $λ$, o…
We give combinatorial descriptions of the terms occurring in continuants of general continued fractions that diverge to three limits. Equating these with the usual combinatorial descriptions due to Euler, Sylvester, and Minding induces nontrivial polynomial identities. Special cases and applications to counting sequences are given.
We give a geometric realization of the polyhedra governed by the structure of associative algebras with co-inner products, or more precisely, governed by directed planar trees. Our explicit realization of these polyhedra, which include the associahedra in a special case, shows in particular that these polyhedra are homeomorphic to balls. We also calculate the number of vertices of the lowest generalized associahedra, giving appropriate generalizations of the Catalan numbers.
Combinatorics of generalized orthogonal polynomials of type $R_{II}$
本文将Kim和Stanton关于类型R_I正交多项式矩的组合模型推广到类型R_{II}，并通过放宽条件定义广义类型R_{II}正交多项式，证明了一个统一多种正交多项式矩组合模型的主定理。
We propose a categorical setting for the study of the combinatorics of rational numbers. We find combinatorial interpretation for the Bernoulli and Euler numbers and polynomials.
Averaging method in combinatorics of symmetric polynomials
本文提出对幂和变量的Schur函数Cauchy恒等式进行平均的方法，该方法类似Borel变换但改变Young图求和的组合因子，为已知恒等式提供统一视角。
An objective of the theory of combinatorial groupoids is to introduce concepts like "holonomy", "parallel transport", "bundles", "combinatorial curvature" etc. in the context of simplicial (polyhedral) complexes, posets, graphs, polytopes, arrangements and other combinatorial objects. In this paper we give an exposition of some of the currently most active research themes in this area, offer a unified point of view, and provide a list of prospective applications in other fields together with a collection of related open problems.
We survey some general-purpose symbolic software packages that implement algorithms from enumerative and analytic combinatorics. Software for the following areas is covered: basic combinatorial objects, symbolic combinatorics, Pólya theory, combinatorial species, and asymptotics. We describe the capabilities that the packages offer as well as some of the algorithms used, and provide links to original documentation. Most of the packages are freely downloadable from the web.
Let $g_1, ..., g_k$ be tropical polynomials in $n$ variables with Newton polytopes $P_1, ..., P_k$. We study combinatorial questions on the intersection of the tropical hypersurfaces defined by $g_1, ..., g_k$, such as the $f$-vector, the number of unbounded faces and (in case of a curve) the genus. Our point of departure is Vigeland's work who considered the special case $k=n-1$ and where all Newton polytopes are standard simplices. We generalize these results to arbitrary $k$ and arbitrary Newton polytopes $P_1, ..., P_k$. This provides new formulas for the number of faces and the genus in t…
Polyhedral combinatorics of bisectors
本文研究任意多面体范数下两点平分线的组合结构，引入二分锥与二分扇概念，证明二维情形下二分扇总存在，并确定若干具体范数下的二分扇，进而给出平分线组合复杂度的界。
There are many different algebraic, geometric and combinatorial objects that one can attach to a complex polynomial with distinct roots. In this article we introduce a new object that encodes many of the existing objects that have previously appeared in the literature. Concretely, for every complex polynomial $p$ with $d$ distinct roots and degree at least 2, we produce a canonical compact planar 2-complex that is a compact metric version of a tiled phase diagram. It has a locally CAT(0) metric that is locally Euclidean away from a finite set of interior points indexed by the critical points o…
Combinatorics of semi-toric degenerations of Schubert varieties in type C
本文利用带自相交的pipe dream为C型辛Gelfand-Tsetlin多面体的Schubert类推广引入组合模型，并证明该推广可由斜有丝分裂算子构造。
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