Francois Descouens
We describe a general algorithm for generating various families of ribbon tableaux and computing their spin polynomials. This algorithm is derived from a new matricial coding. An advantage of this new notation lies in the fact that it permits one to generate ribbon tableaux with skew shapes. This algorithm permits us to compute quickly big LLT polynomials in MuPAD-Combinat.
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
We extend the notion of $k$-ribbon tableaux to the Fibonacci lattice, a differential poset defined by R. Stanley in 1975. Using this notion, we describe an insertion algorithm that takes $k$-colored permutations to pairs of $k$-ribbon Fibonacci tableaux of the same shape, and we demonstrate a color-to-spin property, similar to that described by Shimozono and White for ribbon tableaux. We give an evacuation algorithm which relates the pair of $k$-ribbon Fibonacci tableaux obtained through the insertion algorithm to the pair of $k$-ribbon Fibonacci tableaux obtained using Fomin's growth diagrams…
We give an extension of the classical Schensted correspondence to the case of ribbon tableaux, where ribbons are allowed to be of different sizes. This is done by extending Fomin's growth diagram approach of the classical correspondence between permutations and pairs of standard tableaux of the same shape, in particular by allowing signs in the enumeration. As an application we give a combinatorial proof for the column sums of the character table of the symmetric group.
Inspired by the spin-inversion statistics of Schilling et al. and Haglund et al., we relate the symmetry of ribbon functions to a result of van Leeuwen, and also describe the multiplication of a domino function by a Schur function.
The RSK correspondence generalises the Robinson-Schensted correspondence by replacing permutation matrices by matrices with entries in ${\bf N}$, and standard Young tableaux by semistandard ones. For $r>0$, the Robinson-Schensted correspondence can be trivially extended, using the $r$-quotient map, to one between coloured permutations and pairs of standard $r$-ribbon tableaux built on a fixed $r$-core (the Stanton-White correspondence). This correspondence can also be generalised to arbitrary matrices with entries in ${\bf N}^r$ and pairs of semistandard $r$-ribbon tableaux built on a fixed $r…
Hall-Littlewood functions indexed by rectangular partitions, specialized at primitive roots of unity, can be expressed as plethysms. We propose a combinatorial proof of this formula using A. Schilling's bijection between ribbon tableaux and ribbon rigged configurations.
A Type B Analogue to Ribbon Tableaux
本文引入了 James 和 Kerber 定义的 ribbon tableaux 的移位类似物，给出了移位形状的 $k$-ribbon 填充与 $k$-quotient 之间的双射，并证明其生成函数具有对称、Schur 正和 Schur $Q$ 正等性质。
River ribbon tableaux and symmetric group characters
该研究引入河带表aux，证明对称群的统一特征标界，进而证明Moore与Russell的猜想，并应用证明了Dousse和Féray关于斜形状标准表aux数量的猜想。
Lascoux, Leclerc and Thibon have introduced symmetric functions which are spin and weight generating functions for ribbon tableaux. This article is aimed at studying these `ribbon functions' in analogy with Schur functions. In particular we will describe ribbon Pieri and Murnagham-Nakayama formulae, a ribbon Cauchy identity and an algebra involution which `conjugates' the ribbon functions. We will study these functions in the context of the action of the Heisenberg algebra on the Fock space representation of the quantum affine algebra U_q(sl_n)^, discovered by Kashiwara, Miwa and Stern. We wil…
For any convex preorder on the set of positive roots of affine type A, we classify and construct all associated cuspidal and semicuspidal skew shapes. These combinatorial objects correspond to cuspidal and semicuspidal skew Specht modules for the Khovanov-Lauda-Rouquier algebra of affine type A. Cuspidal skew shapes are ribbons, and we show that every skew shape has a unique ordered tiling by cuspidal ribbons. This tiling data provides an upper bound, in the bilexicographic order on Kostant partitions, for labels of simple factors of Specht modules.
The Wronski map is a finite, PGL_2(C)-equivariant morphism from the Grassmannian Gr(d,n) to a projective space (the projectivization of a vector space of polynomials). We consider the following problem. If C_r < PGL_2(C) is a cyclic subgroup of order r, how may C_r-fixed points are in the in a fibre of the Wronski map over a C_r-fixed point in the base? In this paper, we compute a general answer in terms of r-ribbon tableaux. When r=2, this computation gives the number of real points in the fibre of the Wronski map over a real polynomial with purely imaginary roots. More generally, we can comp…
Crystal graphs, in the sense of Kashiwara, carry a natural monoid structure given by identifying words labelling vertices that appear in the same position of isomorphic components of the crystal. In the particular case of the crystal graph for the $q$-analogue of the special linear Lie algebra $\mathfrak{sl}_{n}$, this monoid is the celebrated plactic monoid, whose elements can be identified with Young tableaux. The crystal graph and the so-called Kashiwara operators interact beautifully with the combinatorics of Young tableaux and with the Robinson--Schensted--Knuth correspondence and so prov…
We make a systematic study of a new combinatorial construction called a dual equivalence graph. We axiomatize these graphs and prove that their generating functions are symmetric and Schur positive. By constructing a graph on ribbon tableaux which we transform into a dual equivalence graph, we give a combinatorial proof of the symmetry and Schur positivity of the ribbon tableaux generating functions introduced by Lascoux, Leclerc and Thibon. Using Haglund's formula for the transformed Macdonald polynomials, this also gives a combinatorial formula for the Schur expansion of Macdonald polynomial…
The connection between the generating functions of various sets of tableaux and the appropriate families of quasisymmetric functions is a significant tool to give a direct analytical proof of some advanced bijective results and provide new combinatorial formulas. In this paper we focus on two kinds of type B Littlewood-Richardson coefficients and derive new formulas using weak composition quasisymmetric functions and Chow's quasisymmetric functions. In the type A case these coefficients give the multiplication table for Schur functions, i.e. the generating functions for classical Young tableau…
Irreducible characters in the symmetric group are of special interest in combinatorics. They can be expressed either combinatorially with ribbon tableaux, or algebraically with contents. In this paper, these two expressions are related in a combinatorial way. We first introduce a fine structure in the famous jeu de taquin called "trace forest", with which we are able to count certain types of ribbon tableaux, leading to a simple bijective proof of a character evaluation formula in terms of contents that dates back to Frobenius (1901). Inspired by this proof, we give an inductive scheme that gi…
Symmetry of the generating function of semistandard oscillating tableaux
本文研究了半标准摆动表的生成函数的对称性，证明其F-正、Schur正及饱和牛顿多面体特性。
Based on Schützenberger's evacuation and a modification of jeu de taquin, we give a bijective proof of an identity connecting the generating function of reverse semistandard Young tableaux with bounded entries with the generating function of all semistandard Young tableaux. This solves Exercise 7.102 b of Richard Stanley's book `Enumerative Combinatorics 2'.
In this paper, we introduce the rhombic alternative tableaux, whose weight generating functions provide combinatorial formulae to compute the steady state probabilities of the two-species ASEP. In the ASEP, there are two species of particles, one heavy and one light, hopping right and left on a one-dimensional finite lattice with open boundaries. Parameters $α$, $β$, and $q$ describe the hopping probabilities. The rhombic alternative tableaux are enumerated by the Lah numbers, which also enumerate certain assemblées of permutations. We describe a bijection between the rhombic alternative table…
An inverted semistandard Young tableau is a row-standard tableau along with a collection of inversion pairs that quantify how far the tableau is from being column semistandard. Such a tableau with precisely $k$ inversion pairs is said to be a $k$-inverted semistandard Young tableau. Building upon earlier work by Fresse and the author, this paper develops generating functions for the numbers of $k$-inverted semistandard Young tableau of various shapes $λ$ and contents $μ$. An easily-calculable generating function is given for the number of $k$-inverted semistandard Young tableau that "standardi…
A specialisation of a transformation formula for multi-dimensional elliptic hypergeometric series is used to provide compact, non-determinantal formulae for the generating function with respect to the major index of standard Young tableaux of skew shapes of the form "staircase minus rectangle".
In this paper we first present summation formulas for $k$-order Eulerian polynomials and $1/k$-Eulerian polynomials. We then present combinatorial expansions of $(c(x)D)^n$ in terms of inversion sequences as well as $k$-Young tableaux, where $c(x)$ is a differentiable function in the indeterminate $x$ and $D$ is the derivative with respect to $x$. We define the $g$-indexes of $k$-Young tableaux and Young tableaux, which have important applications in combinatorics. By establishing some relations between $k$-Young tableaux and standard Young tableaux, we express Eulerian polynomials, second-ord…
Simple Generating Functions for Certain Young Tableaux with Periodic Walls
该研究针对带周期墙的$2\times mn$形杨表计数问题，通过Yamanouchi词转化为格路计数，推导得到简洁的生成函数公式并完成推广，丰富了杨表与格路计数的相关理论。
We exhibit a set of operators on pairs of domino tableaux of the same shape sending them to other such pairs with the same right tableau, in such a way that any two pairs with the same right tableau are conjugate by some composition of the operators. Using these operators we give explicit bases for the isotypic components of a classical Kazhdan-Lusztig left cell in terms of Kazhdan-Lusztig basis vectors.
We develop an algebraic framework for ribbon graphs, revealing symmetry properties of (partial) twisted duality. The original ribbon group action of Ellis-Monaghan and Moffatt restricts self-duality, -petriality, or -triality to the canonical identification of a graph's edges with those of its dual, petrial, or trial, whereas the more natural definition allows any isomorphism. Here we define a new ribbon group action on ribbon graphs, using a semidirect product of the original ribbon group with a permutation group, to take (partial) twists and duals of ribbon graphs while also encoding graph i…
Based on the matrix ansatz of Derrida, Evans, Hakim and Pasquier, we prensent a new way of computing the stationary probability of a state of the asym- metric simple exclusion process (ASEP). Through an insertion algorithm over staircase tableaux, we give a combinatorial proof to the current interpretation of the ASEP by these tableaux of Corteel and Williams. The insertion algorithm induces a recursive structure which implies nice factorised formulas for the generating polynomials of staircase tableaux, as well as a bijection with some coloured inversion tables. In addi- tion, we adapt the in…
Lusztig's fake degree is the generating polynomial for the major index of standard Young tableaux of a given shape. Results of Springer and James & Kerber imply that, mysteriously, its evaluation at a $k$-th primitive root of unity yields the number of border strip tableaux with all strips of size $k$, up to sign. This is essentially the special case of the Murnaghan-Nakayama rule for evaluating an irreducible character of the symmetric group at a rectangular partition. We refine this result to standard Young tableaux and border strip tableaux with a given number of descents. To do so, we intr…
We present a randomized algorithm for generating standard set-valued tableaux by extending the Green-Nijenhuis-Wilf hook walk algorithm. In the case of asymptotically rank two partitions, we use this algorithm to give a fully polynomial almost uniform sampler (FPAUS) for standard set-valued tableaux. This FPAUS is then used to construct a fully polynomial randomized approximation scheme (FPRAS) for counting the number of standard set-valued tableaux for such shapes. We also construct a FPAUS and FPRAS for standard set-valued tableaux when either the size of the partition or the difference betw…
Dumont and Foata introduced in 1976 a three-variable symmetric refinement of Genocchi numbers, which satisfies a simple recurrence relation. A six-variable generalization with many similar properties was later considered by Dumont. They generalize a lot of known integer sequences, and their ordinary generating function can be expanded as a Jacobi continued fraction. We give here a new combinatorial interpretation of the six-variable polynomials in terms of the alternative tableaux introduced by Viennot. A powerful tool to enumerate alternative tableaux is the so-called "matrix Ansatz", and usi…
We present a bijective proof of the hook-length formula for shifted standard tableaux of a fixed shape based on a modified jeu de taquin and the ideas of the bijective proof of the hook-length formula for ordinary standard tableaux by Novelli, Pak and Stoyanovskii. In their proof Novelli, Pak and Stoyanovskii define a bijection between arbitrary fillings of the Ferrers diagram with the integers $1,2,...,n$ and pairs of standard tableaux and hook tabloids. In our shifted version of their algorithm the map from the set of arbitrary fillings of the shifted Ferrers diagram onto the set of shifted …
Straight polyomino tilings of rectangles and special rim-hook tableaux
通过故障线分解与Hadamard积框架，结合Littlewood恒等式和Graham定理，推导了$2k\ imes n$矩形被$k\ imes 1$直线瓷砖覆盖的加权生成函数显式表达式。
In 2012, Sagan and Savage introduced the notion of $st$-Wilf equivalence for a statistic $st$ and for sets of permutations that avoid particular permutation patterns which can be extended to generalized permutation patterns. In this paper we consider $inv$-Wilf equivalence on sets of two or more consecutive permutation patterns. We say that two sets of generalized permutation patterns $Π$ and $Π'$ are $inv$-Wilf equivalent if the generating function for the inversion statistic on the permutations that simultaneously avoid all elements of $Π$ is equal to the generating function for the inversio…
Promotion and cyclic sieving on families of SSYT
该研究针对几类半标准杨表，验证其在提升操作下的循环筛分现象，提出计算斜半标准杨表电荷的新方法，推广了此前相关研究结果。
We describe a generating tree approach to the enumeration and exhaustive generation of k-nonnesting set partitions and permutations. Unlike previous work in the literature using the connections of these objects to Young tableaux and restricted lattice walks, our approach deals directly with partition and permutation diagrams. We provide explicit functional equations for the generating functions, with k as a parameter.
Another bijective proof of Stanley's hook-content formula for the generating function for semistandard tableaux of a given shape is given that does not involve the involution principle of Garsia and Milne. It is the result of a merge of the modified jeu de taquin idea from the author's previous bijective proof (``An involution principle-free bijective proof of Stanley's hook-content formula", Discrete Math. Theoret. Computer Science, to appear) and the Novelli-Pak-Stoyanovskii bijection (Discrete Math. Theoret. Computer Science 1 (1997), 53-67) for the hook formula for standard Young tableaux …
Richardson tableaux and Motzkin paths
本文从莫茨金路径视角研究理查森表，引入独立于RS算法的形状算法，构造局部双射并重新证明q计数公式，还证明了郭关于特定理查森表生成函数的猜想。
This paper concerns a relatively new combinatorial structure called staircase tableaux. They were introduced in the context of the asymmetric exclusion process and Askey--Wilson polynomials, however, their purely combinatorial properties have gained considerable interest in the past few years. In this paper we further study combinatorial properties of staircase tableaux. We consider a general model of staircase tableaux in which symbols that appear in staircase tableaux may have arbitrary positive weights. Under this general model we derive a number of results. Some of our results concern the …
We show that the flagged Grothendieck polynomials defined as the generating functions of flagged set-valued tableaux of Knutson-Miller-Yong can be expressed by a Jacobi-Trudi type determinant formula generalizing the work of Hudson-Matsumura. We also introduce the flagged skew Grothendieck polynomials in these two expressions and show that they coincide.
P{ó}lya urns are urns where at each unit of time a ball is drawn and is replaced with some other balls according to its colour. We introduce a more general model: The replacement rule depends on the colour of the drawn ball and the value of the time (mod p). We discuss some intriguing properties of the differential operators associated to the generating functions encoding the evolution of these urns. The initial partial differential equation indeed leads to ordinary linear differential equations and we prove that the moment generating functions are D-finite. For a subclass, we exhibit a closed…
Integrable systems and crystals for edge labeled tableaux
本文引入边 Schur 函数 $E^λ$，利用可解格点模型证明其对称性并建立 Cauchy 型恒等式，又通过边标记表晶体给出正 Schur 展开及其与去拥挤算法的缠结关系。
We compute the generating function of column-strict plane partitions with parts in {1,2,...,n}, at most c columns, p rows of odd length and k parts equal to n. This refines both, Krattenthaler's ["The major counting of nonintersecting lattice paths and generating functions for tableaux", Mem. Amer. Math. Soc. 115 (1995)] and the author's ["A method for proving polynomial enumeration formulas", preprint] refinement of the Bender-Knuth (ex-)Conjecture. The result is proved by an extension of the method for proving polynomial enumeration formulas which was introduced by the author to q-quasi-poly…
Recently, Gross, Mansour and Tucker introduced the partial-dual genus polynomial of a ribbon graph as a generating function that enumerates the partial duals of the ribbon graph by genus. It is analogous to the extensively-studied polynomial in topological graph theory that enumerates by genus all embeddings of a given graph. To investigate the partial-dual genus polynomial one only needs to focus on bouquets, i.e. ribbon graphs with only one vertex. In this paper, we shall further show that the partial-dual genus polynomial of a bouquet essentially depends on the signed intersection graph of …
Demazure crystals for flagged key polynomials
本文推广关键表与约化分解上的晶体结构，通过标记化弱 EG 插入建立双射和晶体同构，证明标记关键表承载 Demazure 晶体并生成标记关键多项式。
We extend the result of our earlier paper "A family of operators generating domino tableaux..." to type $D$, showing that the same recipe holds for computing basis vectors of isotypic components of Kazhdan-Lusztig cells in that type.
Introducing a vertex polynomial invariant for embedded graphs
针对嵌入图扭曲对偶中的顶点计数问题，提出顶点多项式这一轨道分布生成函数，建立其边界分量等价形式、删边/收缩递归及花束图的带符号相交图依赖，并给出与interlace多项式和拓扑转移多项式相联系的拓扑解释。
On a conjecture of Gross, Mansour and Tucker for $Δ$-matroids
本文研究 Gross、Mansour 和 Tucker 关于 delta-拟阵部分对偶多项式的猜想，证明偶非二元 delta-拟阵均无改变宽度的扭转，扩展了 Yan 和 Jin 对偶正规二元情形的结论。
The symmetric Grothendieck polynomials representing Schubert classes in the $K$-theory of Grassmannians are generating functions for semistandard set-valued tableaux. We construct a type $A_n$ crystal structure on these tableaux. This crystal yields a new combinatorial formula for decomposing symmetric Grothendieck polynomials into Schur polynomials. For single-columns and single-rows, we give a new combinatorial interpretation of Lascoux polynomials (K-analogs of Demazure characters) by constructing a K-theoretic analog of crystals with an appropriate analog of a Demazure crystal. We relate o…
A categorification for the partial-dual genus polynomial
本文利用与无定向拓扑量子场论相关的扩展Frobenius代数，为带状图的部分对偶亏格多项式提供了一种范畴化构造。
Contribution of $n$-cylinder square-tiled surfaces to Masur-Veech volume of $\mathcal{H}(2g-2)$
本文纯组合地推导出 $n$-柱平铺曲面对 $\mathcal{H}(2g-2)$ 的 Masur-Veech 体积贡献的生成函数，通过研究度量带状图计数函数揭示了其与 Kontsevich 多项式的联系。
Combinatorial formulas for shifted dual stable Grothendieck polynomials
本文证明了关于偶 $K$理论 Schur $P$-和 $Q$-函数权生成函数的猜想，推导了其在 $ω$ 对合下的生成函数公式，并验证了 $GQ$-函数构成环基的猜想。
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。