Aart Blokhuis
We describe some combinatorial problems in finite projective planes and indicate how Rédei's theory of lacunary polynomials can be applied to them.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
Some of the classical orthogonal polynomials such as Hermite, Laguerre, Charlier, etc. have been shown to be the generating polynomials for certain combinatorial objects. These combinatorial interpretations are used to prove new identities and generating functions involving these polynomials. In this paper we apply Foata's combinatorial interpretation of the Hermite polynomials as counting matchings of a set to obtain a triple lacunary generating function for the Hermite polynomials. We also give an umbral proof of this generating function.
We prove an infinite family of lacunary recurrences for the Lucas numbers using combinatorial means.
For a sequence $P=(p_n(x))_{n=0}^{\infty}$ of polynomials $p_n(x)$, we study the $K$-tuple and $L$-shifted exponential lacunary generating functions $\mathcal{G}_{K,L}(λ;x):=\sum_{n=0}^{\infty}\frac{λ^n}{n!} p_{n\cdot K+L}(x)$, for $K=1,2\dotsc$ and $L=0,1,2\dotsc$. We establish an algorithm for efficiently computing $\mathcal{G}_{K,L}(λ;x)$ for generic polynomial sequences $P$. This procedure is exemplified by application to the study of Hermite polynomials, whereby we obtain closed-form expressions for $\mathcal{G}_{K,L}(λ;x)$ for arbitrary $K$ and $L$, in the form of infinite series involvi…
Let $F(X) = \sum_{n \geq 0} (-1)^{\varepsilon_n} X^{-λ_n}$ be a real lacunary formal power series, where $\varepsilon_n = 0, 1$ and $λ_{n+1}/λ_n > 2$. It is known that the denominators $Q_n(X)$ of the convergents of its continued fraction expansion are polynomials with coefficients $0, \pm 1$, and that the number of nonzero terms in $Q_n(X)$ is the $n$th term of the Stern-Brocot sequence. We show that replacing the index $n$ by any 2-adic integer $ω$ makes sense. We prove that $Q_ω(X)$ is a polynomial if and only if $ω\in {\mathbb Z}$. In all the other cases $Q_ω(X)$ is an infinite formal powe…
In the paper we obtain some new applications of well--known W. Rudin's theorem concerning lacunary series to problems of combinatorial number theory. We generalize a result of M.-C. Chang on L_2 (L)-norm of Fourier coefficients of a set (here L is a dissociated set), and prove a dual version of the theorem. Our main instrument is computing of eigenvalues of some operators.
We prove a combinatorial formula for the alternating descent polynomials of type A and B. Combining with Josuat-Vergès' combinatorial interpretation for Hoffman's derivative polynomials for tangent and secant functions, we obtain a unified combinatorial proof of Ma-Yeh's two formulae linking these two families of polynomials.
In the case of two combinatorial polynomials, we show that they can exhibited as moments of paramaterized families of orthogonal polynomials, and hence derive their Hankel transforms. Exponential Riordan arrays are the main vehicles used for this.
We prove the positivity of Kazhdan-Lusztig polynomials for sparse paving matroids, which are known to be logarithmically almost all matroids, but are conjectured to be almost all matroids. The positivity follows from a remarkably simple combinatorial formula we discovered for these polynomials using skew young tableaux. This supports the conjecture that Kazhdan-Lusztig polynomials for all matroids have non-negative coeffiecients. In special cases, such as uniform matroids, our formula has a nice combinatorial interpretation.
In this paper, by using some families of special numbers and polynomials with their generating functions, we give various properties of these numbers and polynomials. These numbers are related to the well-known numbers and polynomials, which are the Euler numbers, the Stirling numbers of the second kind, the central factorial numbers and the array polynomials. We also discuss some combinatorial interpretations of these numbers related to the rook polynomials and numbers. Furthermore, we give computation formulas for these numbers and polynomials.
On Combinatorial Properties of the degenerate Krawtchouk Appell polynomials
本文研究了退化Krawtchouk Appell多项式的新类别的组合性质，探讨了其与Stirling数、缩放算子、平移算子及正交Bell多项式之间的联系。
We present combinatorial rules (one theorem and two conjectures) concerning three bases of Z[x1,x2,....]. First, we prove a "splitting" rule for the basis of key polynomials [Demazure '74], thereby establishing a new positivity theorem about them. Second, we introduce an extension of [Kohnert '90]'s "moves" to conjecture the first combinatorial rule for a certain deformation [Lascoux '01] of the key polynomials. Third, we use the same extension to conjecture a new rule for the Grothendieck polynomials [Lascoux-Schutzenberger '82].
In recent years, Alexandersson and others proved combinatorial formulas for the Schur function expansion of the horizontal-strip LLT polynomial $G_λ(x;q)$ in some special cases. We associate a weighted graph $Π$ to $λ$ and we use it to express a linear relation among LLT polynomials. We apply this relation to prove an explicit combinatorial Schur-positive expansion of $G_λ(x;q)$ whenever $Π$ is triangle-free. We also prove that the largest power of $q$ in the LLT polynomial is the total edge weight of our graph.
The Al-Salam-Chihara polynomials are an important family of orthogonal polynomials in one variable $x$ depending on 3 parameters $α$, $β$ and $q$. They are closely connected to a model from statistical mechanics called the partially asymmetric simple exclusion process (PASEP) and they can be obtained as a specialization of the Askey-Wilson polynomials. We give two different combinatorial formulas for the coefficients of the (transformed) Al-Salam-Chihara polynomials. Our formulas make manifest the fact that the coefficients are polynomials in $α$, $β$ and $q$ with positive coefficients.
We construct explicit polynomial realizations of some combinatorial Hopf algebras based on various kind of trees or forests, and some more general classes of graphs, ranging from the Connes-Kreimer algebra to an algebra of labelled forests isomorphic to the Hopf algebra of parking functions, and to a new noncommutative algebra based on endofunctions admitting many interesting subalgebras and quotients.
Partial multivariate Bell polynomials have been defined by E.T. Bell in 1934. These polynomials have numerous applications in Combinatorics, Analysis, Algebra, Probabilities, etc. Many of the formulae on Bell polynomials involve combinatorial objects (set partitions, set partitions in lists, permutations, etc.). So it seems natural to investigate analogous formulae in some combinatorial Hopf algebras with bases indexed by these objects. The algebra of symmetric functions is the most famous example of a combinatorial Hopf algebra. In a first time, we show that most of the results on Bell polyno…
A combinatorial formula for Interpolation Macdonald polynomials
本文给出了插值 Macdonald 多项式的带符号多线队列组合公式，推广了 Macdonald 多项式的多线队列公式。
By using the Poisson formula for resultants and the variants of chip-firing game on graphs, we provide a combinatorial method for computing a class of of resultants, i.e. the characteristic polynomials of the adjacency tensors of starlike hypergraphs including hyperpaths and hyperstars,which are given recursively and explicitly.
Let $ρ$ be a non-negative integer. A $ρ$-removed uniform matroid is a matroid obtained from a uniform matroid by removing a collection of $ρ$ disjoint bases. We present a combinatorial formula for Kazhdan-Lusztig polynomials of $ρ$-removed uniform matroids, using skew Young Tableaux. Even for uniform matroids, our formula is new, gives manifestly positive integer coefficients, and is more manageable than known formulas.
The incomplete tribonacci polynomials, denoted by T_n^{(s)}(x), generalize the usual tribonacci polynomials T_n(x) and were introduced in [10], where several algebraic identities were shown. In this paper, we provide a combinatorial interpretation for T_n^{(s)}(x) in terms of weighted linear tilings involving three types of tiles. This allows one not only to supply combinatorial proofs of the identities for T_n^{(s)}(x) appearing in [10] but also to derive additional identities. In the final section, we provide a formula for the ordinary generating function of the sequence T_n^{(s)}(x) for a f…
Local obstructions in sequences revisited
本文通过引入一个简单的组合游戏及其必胜策略，统一证明了非重复序列和缺项序列分母逼近的已知结果，简化了证明并改进了界限，同时得到了可计算版本。
This paper presents the first combinatorial polynomial-time algorithm for minimizing submodular set functions, answering an open question posed in 1981 by Grotschel, Lovasz, and Schrijver. The algorithm employs a scaling scheme that uses a flow in the complete directed graph on the underlying set with each arc capacity equal to the scaled parameter. The resulting algorithm runs in time bounded by a polynomial in the size of the underlying set and the largest length of the function value. The paper also presents a strongly polynomial-time version that runs in time bounded by a polynomial in the…
A barred preferential arrangement is a preferential arrangement, onto which in-between the blocks of the preferential arrangement a number of identical bars are inserted. We offer a generalisation of barred preferential arrangements by making use of the generalised Stirling numbers proposed by Hsu and Shiue (1998). We discuss how these generalised barred preferential arrangements offer a unified combinatorial interpretation of geometric polynomials. We also discuss asymptotic properties of these numbers.
We give a combinatorial expansion of the stable Grothendieck polynomials of skew Young diagrams in terms of skew Schur functions, using a new row insertion algorithm for set-valued semistandard tableaux of skew shape. This expansion unifies some previous results: it generalizes a combinatorial formula obtained in earlier joint work with López Martín and Teixidor i Bigas concerning Brill-Noether curves, and it generalizes a 2000 formula of Lenart and a recent result of Reiner-Tenner-Yong to skew shapes. We also give an expansion in the other direction: expressing skew Schur functions in terms o…
In this note we generalize the convolution formula for the Tutte polynomial of Kook-Reiner-Stanton and Etienne-Las Vergnas to a more general setting that includes both arithmetic matroids and delta-matroids. As corollaries, we obtain new proofs of two positivity results for pseudo-arithmetic matroids and a combinatorial interpretation of the arithmetic Tutte polynomial at infinitely many points in terms of arithmetic flows and colorings. We also exhibit connections with a decomposition of Dahmen-Micchelli spaces and lattice point counting in zonotopes.
We present a decomposition of the generalized binomial coefficients associated with Jack polynomials into two factors: a stem, which is described explicitly in terms of hooks of the indexing partitions, and a leaf, which inherits various recurrence properties from the binomial coefficients and depends exclusively on the skew diagram. We then derive a direct combinatorial formula for the leaf in the special case where the two indexing partitions differ by at most two rows. This formula also exhibits an unexpected symmetry with respect to the lengths of the two rows.
Using the combinatorial formula for the transformed Macdonald polynomials of Haglund, Haiman, and Loehr, we investigate the combinatorics of the symmetry relation $\widetilde{H}_μ(\mathbf{x};q,t) = \widetilde{H}_{μ^\ast}(\mathbf{x};t,q)$. We provide a purely combinatorial proof of the relation in the case of Hall-Littlewood polynomials ($q=0$) when $μ$ is a partition with at most three rows, and for the coefficients of the square-free monomials in $\mathbf{x}$ for all shapes $μ$. We also provide a proof for the full relation in the case when $μ$ is a hook shape, and for all shapes at the speci…
We consider the symmetric group $S_n$-module of the polynomial ring with $m$ sets of $n$ commuting variables and $m'$ sets of $n$ anti-commuting variables and show that the multiplicity of an irreducible indexed by the partition $λ$ (a partition of $n$) is the number of multiset tableaux of shape $λ$ satisfying certain column and row strict conditions. We also present a finite generating set for the ring of $S_n$ invariant polynomials of this ring.
Algebraic and Combinatorial Stability of Independence Polynomials in Iterated Strong Products of Cycles
本文通过代数和组合方法研究了迭代强积循环图中独立多项式的稳定性，揭示了其对数凹性和单峰性特性，并探讨了其在图论和统计力学中的应用。
We consider two families of polynomials that play the same role in the Temperley Lieb algebra of a Coxeter group as the Kazhdan Lusztig and R polynomials play in the Hecke algebra of the group. We study these polynomials from a combinatorial point of view. More precisely we obtain recursions, non recursive formulas, symmetry properties, and expressions for the constant terms, of these polynomials.
Combinatorial relations among restricted and half Eulerian polynomials of types $A$, $B$, and $D$
本文研究了类型 A、B 和 D 的欧拉多项式之间的组合关系，推导了多项恒等式并解决了开放性问题。
Combinatorial interpretation of the coefficients of the order polynomial of fence posets
研究针对围栏偏序集定义新统计量blP，为其序多项式系数提供组合解释，还借此解释舒伯特拟阵基多面体埃尔哈特多项式系数，建立序多面体埃尔哈特多项式线性系数下界，并猜想该统计量对其他偏序集的推广。
Two combinatorial puzzles arising from the theory of Kohnert polynomials
本文研究由Kohnert多项式理论引出的两个组合谜题：给定初始图，分别用最少或最多Kohnert移动构造固定图，并给出完整解及移动次数的组合计算方法。
Grothendieck polynomials of inverse fireworks permutations
本文通过引入可重构管道梦想公式的新组合对象，得到格罗滕迪克多项式最高次分量的首个直接组合公式，并证明了相关猜想的逆烟花情形。
Combinatorial $t$-designs have nice applications in coding theory, finite geometries and several engineering areas. There are two major methods of constructing $t$-designs. One of them is via group actions of certain permutation groups which are $t$-transitive or $t$-homogeneous on some point set. The other is a coding-theoretical one. The objectives of this paper are to introduce two constructions of $t$-designs with special polynomials over finite fields GF$(q)$, and obtain $2$-designs and $3$-designs with interesting parameters. A type of d-polynomials is defined and used to construct $2$-d…
Let G be a graph, and let $χ$G be its chromatic polynomial. For any non-negative integers i, j, we give an interpretation for the evaluation $χ$ (i) G (--j) in terms of acyclic orientations. This recovers the classical interpretations due to Stanley and to Green and Zaslavsky respectively in the cases i = 0 and j = 0. We also give symmetric function refinements of our interpretations, and some extensions. The proofs use heap theory in the spirit of a 1999 paper of Gessel.
Bijections between different combinatorial models for $q$-Whittaker and modified Hall-Littlewood polynomials
本文为$q$-Whittaker和修正Hall-Littlewood多项式的两种组合公式所诱导的参数集构造保持内容和主指标的双射，并借助charge/cocharge理论证明其正确性。
We prove some functional equations involving the (classical) matching polynomials of path and cycle graphs and the $d$-matching polynomial of a cycle graph. A matching in a (finite) graph $G$ is a subset of edges no two of which share a vertex, and the matching polynomial of $G$ is a generating function encoding the numbers of matchings in $G$ of each size. The $d$-matching polynomial is a weighted average of matching polynomials of degree-$d$ covers, and was introduced in a paper of Hall, Puder, and Sawin. Let $\mathcal{C}_n$ and $\mathcal{P}_n$ denote the respective matching polynomials of t…
We give a combinatorial proof that the product of a Schubert polynomial by a Schur polynomial is a nonnegative sum of Schubert polynomials. Our proof uses Assaf's theory of dual equivalence to show that a quasisymmetric function of Bergeron and Sottile is Schur-positive. By a geometric comparison theorem of Buch and Mihalcea, this implies the nonnegativity of Gromov-Witten invariants of the Grassmannian.
We give a new characterization of the Tutte polynomial of graphs. Our characterization is formally close (but inequivalent) to the original definition given by Tutte as the generating function of spanning trees counted according to activities. Tutte's notion of activity requires to choose a \emph{linear order} on the edge set (though the generating function of the activities is, in fact, independent of this order). We define a new notion of activity, the embedding-activity, which requires to choose a \emph{combinatorial embedding} of the graph, that is, a cyclic order of the edges around each …
New combinatorial proof of Gaussian polynomial and the monotonicity of the Garvan's $k$-rank
本文通过构造高斯多项式上的新双射，细化算法 Z，并给出广义 Rogers-Ramanujan 恒等式及 Garvan 的 $k$-秩单调性的组合证明。
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。
Charlier configurations provide a combinatorial model for Charlier polynomials. We use this model to give a combinatorial proof of a multilinear generating function for Charlier polynomials. As special cases of the multilinear generating function, we obtain the bilinear generating function for Charlier polynomials and formulas for derangements.
Polynomial and combinatorial analogues of Gauss congruence
本文加强了$q$-Gauss同余与基于循环群作用的Gauss同余组合类比之间的联系，通过显式公式刻画$q$-Gauss同余，并基于可变颜色和长度的项链开发通用组合模型，产生多个循环筛分新例子，同时将Gauss同余扩展至任意赋秩半群索引的序列。
On a combinatorial puzzle arising from the theory of Lascoux polynomials
本文研究将Lascoux多项式定义中的密钥图替换为任意图时产生的组合谜题，给出了多种图族及贪心策略下一般图的最大幽灵方格数计算方法。
We give combinatorial formulas for F-polynomials in cluster algebras of classical types in terms of the weighted paths in certain directed graphs. As a consequence we prove the positivity of F-polynomials in cluster algebras of classical types.
Using purely combinatorial means we obtain results on simultaneous Diophantine approximation modulo 1 for systems of polynomials with real coefficients and no constant term.
We give a combinatorial formula for the non-symmetric Macdonald polynomials E_μ(x;q,t). The formula generalizes our previous combinatorial interpretation of the integral form symmetric Macdonald polynomials J_μ(x;q,t). We prove the new formula by verifying that it satisfies a recurrence, due to Knop, that characterizes the non-symmetric Macdonald polynomials.
In this article, we introduce combinatorial models for poly-Bernoulli polynomials and poly-Euler numbers of both kinds. As their applications, we provide combinatorial proofs of some identities involving poly-Bernoulli polynomials.
We propose a combinatorial interpretation of the coefficient of $q$ in Kazhdan- Lusztig polynomials and we prove it for finite simply-laced Weyl groups.
Combinatorial $B_n$-analogues of Schubert polynomials
Schubert多项式的组合$B_n$ -类似物和相应的对称函数是由包含超八面体群的nilCoxeter代数的$B_n$ -Yang-Baxter方程的指数解构造的.
We prove a combinatorial formula for the Macdonald polynomial H_mu(x;q,t) which had been conjectured by the first author. Corollaries to our main theorem include the expansion of H_mu(x;q,t) in terms of LLT polynomials, a new proof of the charge formula of Lascoux and Schutzenberger for Hall-Littlewood polynomials, a new proof of Knop and Sahi's combinatorial formula for Jack polynomials as well as a lifting of their formula to integral form Macdonald polynomials, and a new combinatorial rule for the Kostka-Macdonald coefficients K_{lambda,mu}(q,t) in the case that mu is a partition with parts…
Combinatorial spectra using polynomials
本文提出基于多项式环上组合谱的新方法，用于研究图的幻型着色和支配问题，旨在为图论学家提供潜在有用的工具。
Applying techniques similar to Combinatorial Nullstellensatz we prove a lower estimate of $|f(A,B)|$ for finite subsets $A$, $B$ of a field, and polynomial $f(x,y)$ of the form $f(x,y)=g(x)+yh(x)$, where degree of $g$ is greater then degree of $h$.
We describe a classification of degree n complex coefficient polynomials with respect to combinatorial patterns that arise from the two real algebraic curves obtained as the zero sets for their real and imaginary part. In particular, we work out explicitly this classification for degree 3 polynomials, and other special families of polynomials. This work extends to the singular case similar considerations of Martin, Savitt, and Singer for non-singular basketballs.
In 2017 Aguiar and Ardila provided a generic way to construct polynomial invariants of combinatorial objects using the notions of Hopf monoids and characters of Hopf monoids. They show that it is possible to find a combinatorial interpretation of these polynomials over negative integers using the antipode and give a cancellation-free grouping-free formula for the antipode on generalized permutahedra. In this work, we give a combinatorial interpretation of these polynomials over both positive integers and negative integers for the Hopf monoids of generalized permutahedra and hypergraphs and for…
In this paper, we present a simple combinatorial proof of a Weyl type formula for hook Schur polynomials, which has been obtained by using a Kostant type cohomology formula for $\frak{gl}_{m|n}$. In general, we can obtain in a combinatorial way a Weyl type formula for various highest weight representations of a Lie superalgebra, which together with a general linear algebra forms a Howe dual pair.
Schubert polynomials are a basis for the polynomial ring that represent Schubert classes for the flag manifold. In this paper, we introduce and develop several new combinatorial models for Schubert polynomials that relate them to other known bases including key polynomials and fundamental slide polynomials. We unify these and existing models by giving simple bijections between the combinatorial objects indexing each. In particular, we give a simple bijective proof that the balanced tableaux of Edelman and Greene enumerate reduced expressions and a direct combinatorial proof of Kohnert's algori…
We develop a marking system for an analog of Hasse diagrams of intervals $[u,v]$ with $u\leq v$ in a Hermitian symmetric pair $W/W_J$, and use this to create a closed form algorithm for computing relative R-polynomials. The uniform nature of this algorithm allows us to show combinatorial invariance of relative Kazhdan-Lusztig polynomials in the Hermitian symmetric space setting.
I give a combinatorial interpretation of the multiple Laguerre polynomials of the first kind of type II, generalizing the digraph model found by Foata and Strehl for the ordinary Laguerre polynomials. I also give an explicit integral representation for these polynomials, which shows that they form a multidimensional Stieltjes moment sequence whenever $x \le 0$.
Flipclasses and Combinatorial Invariance for Kazhdan--Lusztig polynomials
本文通过引入翻转类概念构造对称群区间的组合不变量，给出计算Kazhdan--Lusztig $\widetilde{R}$-多项式低次项系数的配方，为组合不变性猜想提供新证据。
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。