A `nice' bijection for a content formula for skew semistandard Young tableaux

Pieri rules for classical groups and equinumeration between generalized oscillating tableaux and semistandard tableaux

We present several equinumerous results between generalized oscillating tableaux and semistandard tableaux and give a representation-theoretical proof to them. As one of the key ingredients of the proof, we provide Pieri rules for the symplectic and orthogonal groups.

极小斜半标准表与Hillman--Grassl对应

Minimal skew semistandard tableaux and the Hillman--Grassl correspondence

本文定义极小斜半标准表，通过新的Hillman--Grassl双射描述与激发图对应，并证明NHLF加数少于OOF加数，刻画两者加数相等的形状。

Inversions of Semistandard Young Tableaux

A tableau inversion is a pair of entries from the same column of a row-standard tableau that lack the relative ordering necessary to make the tableau column-standard. An $i$-inverted Young tableau is a row-standard tableau with precisely $i$ inversion pairs, and may be interpreted as a generalization of (column-standard) Young tableau. Inverted Young tableau that lack repeated entries were introduced by Fresse to calculate the Betti numbers of Springer fibers in Type A, and were later developed as combinatorial objects in their own right by Beagley and Drube. This paper generalizes earlier not…

晶体与符号反转对合：集值斜杨表和五顶点模型的装饰态

Crystal operators and a sign-reversing involution on semistandard set-valued tableaux of skew shape

本文构造了集值斜杨表、标记Gelfand-Tsetlin模式与五顶点模型装饰态之间的双射，保持权重并交织晶体算子，证明了纤维的布尔格结构及符号反转对合的性质。

具有有理特征步长的框架与半标准杨表之间的联系

Connections Between Frames with Rational Eigensteps and Semistandard Young Tableaux

本研究建立了有理特征步长框架与半标准杨表的对应关系，提出基于杨表的框架特征步长构造新方法，并用组合技术实现框架补的对应，为框架分析提供组合学新视角。

Semistandard Tableaux for Demazure Characters (Key Polynomials) and Their Atoms

The Schur function indexed by a partition lambda with at most n parts is the sum of the weight monomials for the Young tableaux of shape lambda. Let pi be an n-permutation. We give two descriptions of the tableaux that contribute their monomials to the key polynomial indexed by pi and lambda. (These polynomials are the characters of the Demazure modules for GL(n).) The "atom" indexed by pi is the sum of weight monomials of the tableaux whose right keys are the "key" tableau for pi. Schur functions and key polynomials can be decomposed into sums of atoms. We also describe the tableaux that cont…

A cornucopia of quasi-Yamanouchi tableaux

Quasi-Yamanouchi tableaux are a subset of semistandard Young tableaux and refine standard Young tableaux. They are closely tied to the descent set of standard Young tableaux and were introduced by Assaf and Searles to tighten Gessel's fundamental quasisymmetric expansion of Schur functions. The descent set and descent statistic of standard Young tableaux repeatedly prove themselves useful to consider, and as a result, quasi-Yamanouchi tableaux make appearances in many ways outside of their original purpose. Some examples, which we present in this paper, include the Schur expansion of Jack poly…

Promotion and Cyclic Sieving on Rectangular $δ$-Semistandard Tableaux

Let $δ=(δ_1,\ldots,δ_n)$ be a string of letters $h$ and $v$. We define a Young tableau to be $δ$-semistandard if the entries are weakly increasing along rows and columns, and the entries $i$ form a horizontal strip if $δ_i=h$ and a vertical strip if $δ_i=v$. We define $δ$-promotion on such tableaux via a modified jeu-de-taquin. The first main result is that $δ$-promotion has period $n$ on rectangular $δ$-semistandard tableaux, generalizing the results of Haiman and Rhoades for standard and semistandard tableaux. The second main result states that the set of rectangular $δ$-semistandard tableau…

Crystals, semistandard tableaux and cyclic sieving phenomenon

In this paper, we study a new cyclic sieving phenomenon on the set $\mathsf{SST}_n(λ)$ of semistandard Young tableaux with the cyclic action $\mathsf{c}$ arising from its $U_q(\mathfrak{sl}_n)$-crystal structure. We prove that if $λ$ is a Young diagram with $\ell(λ) < n$ and $\gcd( n, |λ| )=1$, then the triple $\left( \mathsf{SST}_n(λ), \mathsf{C}, q^{- κ(λ)} s_λ(1,q, \ldots, q^{n-1}) \right) $ exhibits the cyclic sieving phenomenon, where $\mathsf{C}$ is the cyclic group generated by $\mathsf{c}$. We further investigate a connection between $\mathsf{c}$ and the promotion $\mathsf{pr}$ and sho…

$\mathfrak{q}$-crystal structure on primed tableaux and on signed unimodal factorizations of reduced words of type $B$

Crystal basis theory for the queer Lie superalgebra was developed by Grantcharov et al. and it was shown that semistandard decomposition tableaux admit the structure of crystals for the queer Lie superalgebra or simply $\mathfrak{q}$-crystal structure. In this paper, we explore the $\mathfrak{q}$-crystal structure of primed tableaux (semistandard marked shifted tableaux) and that of signed unimodal factorizations of reduced words of type $B$. We give the explicit odd Kashiwara operators on primed tableaux and the forms of the highest and lowest weight vectors. We also give the explicit algorit…

New Descriptions of Demazure Tableaux and Right Keys, with Applications to Convexity

The right key of a semistandard Young tableau is a tool used to find Demazure characters for $sl_n(\mathbb{C})$. This thesis gives methods to obtain the right and left keys by inspection of the semistandard Young tableau. Given a partition $λ$ and a Weyl group element $w$, there is a semistandard Young tableau $Y_λ(w)$ of shape $λ$ that corresponds to $w$. The Demazure character for $λ$ and $w$ is known to be the sum of the weights of all tableaux whose right key is dominated by $Y_λ(w)$. The set of all such tableaux is denoted $\mathcal{D}_λ(w)$. Exploiting the method mentioned above for obta…

An explicit bijection between semistandard tableaux and non-elliptic sl_3 webs

The sl_3 spider is a diagrammatic category used to study the representation theory of the quantum group U_q(sl_3). The morphisms in this category are generated by a basis of non-elliptic webs. Khovanov- Kuperberg observed that non-elliptic webs are indexed by semistandard Young tableaux. They establish this bijection via a recursive growth algorithm. Recently, Tymoczko gave a simple version of this bijection in the case that the tableaux are standard and used it to study rotation and joins of webs. We build on Tymoczko's bijection to give a simple and explicit algorithm for constructing all no…

Set-valued domino tableaux and shifted set-valued domino tableaux

We prove K-theoretic and shifted K-theoretic analogues of the bijection of Stanton and White between domino tableaux and pairs of semistandard tableaux. As a result, we obtain product formulas for pairs of stable Grothendieck polynomials and pairs of K-theoretic Q-Schur functions.

Set-Valued Skyline Fillings

Set-valued tableaux play an important role in combinatorial $K$-theory. Separately, semistandard skyline fillings are a combinatorial model for Demazure atoms and key polynomials. We unify these two concepts by defining a set-valued extension of semistandard skyline fillings and then give analogues of results of J. Haglund, K. Luoto, S. Mason, and S. van Willigenberg. Additionally, we give a bijection between set-valued semistandard Young tableaux and C. Lenart's Schur expansion of the Grothendieck polynomial $G_λ$, using the uncrowding operator of V. Reiner, B. Tenner, and A. Yong.

关于设定值表格的奇数性的一种递归方法

A recursive method for the oddness of the number of set-valued tableaux

本文提出了一种递归方法，证明给定形状的设定值表格数量为奇数。

Prism tableaux for alternating sign matrix varieties

A prism tableau is a set of reverse semistandard tableaux, each positioned within an ambient grid. Prism tableaux were introduced to provide a formula for the Schubert polynomials of A. Lascoux and M.P. Schützenberger. This formula directly generalizes the well known expression for Schur polynomials as a sum over semistandard tableaux. Alternating sign matrix varieties generalize the matrix Schubert varieties of W. Fulton. We use prism tableaux to give a formula for the multidegree of an alternating sign matrix variety.

Permuted composition tableaux, 0-Hecke algebra and labeled binary trees

We introduce a generalization of semistandard composition tableaux called permuted composition tableaux. These tableaux are intimately related to permuted basement semistandard augmented fillings studied by Haglund, Mason and Remmel. Our primary motivation for studying permuted composition tableaux is to enumerate all possible ordered pairs of permutations $(σ_1,σ_2)$ that can be obtained by standardizing the entries in two adjacent columns of an arbitrary composition tableau. We refer to such pairs as compatible pairs. To study compatible pairs in depth, we define a $0$-Hecke action on permut…

Crystal graphs for shifted tableaux

We define crystal operators on semistandard shifted tableaux, giving a new proof that Schur $P$-functions are Schur positive. We define a queer crystal operator to construct a connected queer crystal on semistandard shifted tableaux of a given shape, providing a new proof that products of Schur $P$-functions are Schur $P$-positive. We also give a rectification map from shifted tableaux to Young tableaux that commutes with the crystal operators and provides a dual algorithm to shifted insertion.

Combinatorial proof of the skew K-saturation theorem

We give a combinatorial proof of the skew Kostka analogue of the K-saturation theorem. More precisely, for any positive integer k, we give an explicit injection from the set of skew semistandard Young tableaux with skew shape kλ/kμ and type kν to the set of skew semistandard Young tableaux of shape λ/μ and type ν. Based on this method, we pose some natural conjectural refinements on related problems.

Hook-content formulae for symplectic and orthogonal tableaux

By considering the specialisation $s_λ(1,q,q^2,...,q^{n-1})$ of the Schur function, Stanley was able to describe a formula for the number of semistandard Young tableaux of shape $λ$ in terms of two properties of the boxes in the diagram for $λ$. Using specialisations of symplectic and orthogonal Schur functions, we derive corresponding formulae, first given by El Samra and King, for the number of semistandard symplectic and orthogonal $λ$-tableaux.

Some bijective correspondences involving domino tableaux

We define a number of new combinatorial operations on skew semistandard domino tableaux, which together with constructions introduced earlier by C. Carre and B. Leclerc, define an elegant structure on the set of these tableaux, that closely parallels the structure defined in [math.CO/9908099] for ordinary skew semistandard tableaux, that is related to the Littlewood-Richardson rule. These operations are: (1) a bijection between semistandard domino tableaux and certain pairs of ordinary tableaux of the same weight that together fill the same shape; (2) a weight preserving transformation of domi…

Enumerating quasi-Yamanouchi tableaux of Durfee size two

Quasi-Yamanouchi tableaux connect the two most studied types of tableaux. They are a subset of semistandard Young tableaux that are also a refinement on standard Young tableaux, and they can be used to improve the fundamental quasisymmetric expansion of Schur polynomials. We prove a product formula for enumerating certain quasi-Yamanouchi tableaux and provide strong evidence that no product formula exists in general for other shapes. Along the way, we also prove some nice properties of their distribution and symmetry.

Genomic Tableaux

We explain how genomic tableaux [Pechenik-Yong '15] are a semistandard complement to increasing tableaux [Thomas-Yong '09]. From this perspective, one inherits genomic versions of jeu de taquin, Knuth equivalence, infusion and Bender-Knuth involutions, as well as Schur functions from (shifted) semistandard Young tableaux theory. These are applied to obtain new Littlewood-Richardson rules for K-theory Schubert calculus of Grassmannians (after [Buch '02]) and maximal orthogonal Grassmannians (after [Clifford-Thomas-Yong '14], [Buch-Ravikumar '12]). For the unsolved case of Lagrangian Grassmannia…

斜半标准Young表上的自由作用与循环筛分

Free action and cyclic sieving on skew semi-standard Young tableaux

该研究给出半标准Young表循环筛分现象相关定理的简短证明，并将结果推广到斜形状情形。

Proof of a Conjecture of Reiner-Tenner-Yong on Barely Set-valued Tableaux

The notion of a barely set-valued semistandard Young tableau was introduced by Reiner, Tenner and Yong in their study of the probability distribution of edges in the Young lattice of partitions. Given a partition $λ$ and a positive integer $k$, let ${\mathrm{BSSYT}}(λ,k)$ (respectively, ${\mathrm{SYT}}(λ,k)$) denote the set of barely set-valued semistandard Young tableaux (respectively, ordinary semistandard Young tableaux) of shape $λ$ with entries in row $i$ not exceeding $k+i$. In the case when $λ$ is a rectangular staircase partition $δ_d(b^a)$, Reiner, Tenner and Yong conjectured that $|{…

Shifted tableaux and products of Schur's symmetric functions

We introduce a new combinatorial object, semistandard increasing decomposition tableau and study its relation to a semistandard decomposition tableau introduced by Kraśkiewicz and developed by Lam and Serrano. We also introduce generalized Littlewood--Richardson coefficients for products of Schur's symmetric functions and give combinatorial descriptions in terms of tableau words. The insertion algorithms play central roles for proofs. A new description of shifted Littlewood--Richardson coefficients is given in terms of semistandard increasing decomposition tableaux. We show that a "big" Schur …

Bijections between oscillating tableaux and (semi)standard tableaux via growth diagrams

We prove that the number of oscillating tableaux of length $n$ with at most $k$ columns, starting at $\emptyset$ and ending at the one-column shape $(1^m)$, is equal to the number of standard Young tableaux of size~$n$ with $m$ columns of odd length, all columns of length at most $2k$. This refines a conjecture of Burrill, which it thereby establishes. We prove as well a "Knuth-type" extension stating a similar equi-enumeration result between generalised oscillating tableaux and semistandard tableaux.

Sign matrix polytopes from Young tableaux

Motivated by the study of polytopes formed as the convex hull of permutation matrices and alternating sign matrices, we define several new families of polytopes as convex hulls of sign matrices, which are certain {0,1,-1}-matrices in bijection with semistandard Young tableaux. We investigate various properties of these polytopes, including their inequality descriptions, vertices, facets, and face lattices, as well as connections to alternating sign matrix polytopes and transportation polytopes.

A bijection between $K$-Kohnert diagrams and reverse set-valued tableaux

Lascoux polynomials are $K$-theoretic analogues of the key polynomials. They both have combinatorial formulas involving tableaux: reverse set-valued tableaux ($\mathsf{RSVT}$) rule for Lascoux polynomials and reverse semistandard Young tableaux ($\mathsf{RSSYT}$) rule for key polynomials. Furthermore, key polynomials have a simple algorithmic model in terms of Kohnert diagrams, which are in bijection with $\mathsf{RSSYT}$. Ross and Yong introduced $K$-Kohnert diagrams, which are analogues of Kohnert diagrams. They conjectured a $K$-Kohnert diagram rule for Lascoux polynomials. We establish thi…

A crystal-like structure on shifted tableaux

We introduce coplactic raising and lowering operators $E'_i$, $F'_i$, $E_i$, and $F_i$ on shifted skew semistandard tableaux. We show that the primed operators and unprimed operators each independently form type A Kashiwara crystals (but not Stembridge crystals) on the same underlying set and with the same weight functions. When taken together, the result is a new kind of `doubled crystal' structure that recovers the combinatorics of type B Schubert calculus: the highest-weight elements of our crystals are precisely the shifted Littlewood-Richardson tableaux, and their generating functions are…

Shifted tableaux crystals

We introduce coplactic raising and lowering operators $E'_i$, $F'_i$, $E_i$, and $F_i$ on shifted skew semistandard tableaux. We show that the primed operators and unprimed operators each independently form type A Kashiwara crystals (but not Stembridge crystals) on the same underlying set and with the same weight functions. When taken together, the result is a new kind of `doubled crystal' structure that recovers the combinatorics of type B Schubert calculus: the highest-weight elements of our crystals are precisely the shifted Littlewood-Richardson tableaux, and their generating functions are…

Lenart's bijection via bumpless pipe dreams

Pipe dreams and bumpless pipe dreams for vexillary permutations are each known to be in bijection with certain semistandard tableaux via maps due to Lenart and Weigandt, respectively. Recently, Gao and Huang have defined a bijection between the former two sets. In this note we show for vexillary permutations that the Gao-Huang bijection preserves the associated tableaux, giving a new proof of Lenart's result. Our methods extend to give a recording tableau for any bumpless pipe dream.

半标准杨表族的提升与循环筛分

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。

Lecture hall tableaux

We introduce lecture hall tableaux, which are fillings of a skew Young diagram satisfying certain conditions. Lecture hall tableaux generalize both lecture hall partitions and anti-lecture hall compositions, and also contain reverse semistandard Young tableaux as a limit case. We show that the coefficients in the Schur expansion of multivariate little $q$-Jacobi polynomials are generating functions for lecture hall tableaux. Using a Selberg-type integral we show that moments of multivariate little $q$-Jacobi polynomials, which are equal to generating functions for lecture hall tableaux of a Yo…

Bijecting hidden symmetries for skew staircase shapes

We present a bijection between the set of standard Young tableaux of staircase minus rectangle shape, and the set of marked shifted standard Young tableaux of a certain shifted shape. Numerically, this result is due to DeWitt (2012). Combined with other known bijections this gives a bijective proof of the product formula for the number of standard Young tableaux of staircase minus rectangle shape. This resolves an open problem by Morales, Pak and Panova (2019), and allows for efficient random sampling. Other applications include a bijection for semistandard Young tableaux, and a bijective proo…

关于斜形状标准杨表的 Okounkov-Olshanski 公式

On the Okounkov-Olshanski formula for standard tableaux of skew shapes

本文研究了斜形状标准杨表的 Okounkov-Olshanski 公式，证明了其行列式公式、与其他斜表公式的等价性及 q-模拟，并改进了厚锯齿形状标准杨表数量的上界。

A determinant-like formula for the Kostka numbers

Young tableaux are ubiquitous in various branches of mathematics. There are two counting formulas for standard Young tableaux. The first involves a determinant and goes back to Frobenius and Young, and the second is the hook formula by Frame, Robinson and Thrall. We present a generalization of the determinant formula for semistandard Young tableaux of given shape and of given content. Our counting formula -- though not a determinant -- is a generalization of the determinant formula by Frobenius and Young.

多重射影 Seshadri 分层与 Young 表

Multiprojective Seshadri stratifications and Young-tableaux

本文通过将 Seshadri 分层推广至多重射影簇，为经典半标准 Young 表提供了代数几何解释，揭示了其与 Schubert 簇上有理函数消失重数的对应关系。

Plethysms of symmetric functions and highest weight representations

Let $s_ν\circ s_μ$ denote the plethystic product of the Schur functions $s_ν$ and $s_μ$. In this article we define an explicit polynomial representation corresponding to $s_ν\circ s_μ$ with basis indexed by certain `plethystic' semistandard tableaux. Using these representations we prove generalizations of four results on plethysms due to Bruns--Conca--Varbaro, Brion, Ikenmeyer and the authors. In particular, we give a sufficient condition for the multiplicity $\langle s_ν\circ s_μ, s_λ\rangle$ to be stable under insertion of new parts into $μ$ and $λ$. We also characterize all maximal and mini…

Truncations of random unitary matrices and Young tableaux

Let $U$ be a matrix chosen randomly, with respect to Haar measure, from the unitary group $U(d).$ We express the moments of the trace of any submatrix of $U$ as a sum over partitions whose terms count certain standard and semistandard Young tableaux. Using this combinatorial interpretation, we obtain a simple closed form for the moments of an individual entry of a random unitary matrix and use this to deduce that the entries converge in moments to standard complex Gaussian random variables. In addition, we recover a well-known theorem of E. Rains which shows that the moments of the trace of a …

Proof of a conjecture of Morales-Pak-Panova on reverse plane partitions

Using equivariant cohomology theory, Naruse obtained a hook length formula for the number of standard Young tableaux of skew shape $λ/μ$. Morales, Pak and Panova found two $q$-analogues of Naruse's formula respectively by counting semistandard Young tableaux of shape $λ/μ$ and reverse plane partitions of shape $λ/μ$. When $λ$ and $μ$ are both staircase shape partitions, Morales, Pak and Panova conjectured that the generating function of reverse plane partitions of shape $λ/ μ$ can be expressed as a determinant whose entries are related to $q$-analogues of the Euler numbers. The objective of th…

Another involution principle-free bijective proof of Stanley's hook-content formula

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 …

Combinatorics of patience sorting monoids

This paper makes a combinatorial study of the two monoids and the two types of tableaux that arise from the two possible generalizations of the Patience Sorting algorithm from permutations (or standard words) to words. For both types of tableaux, we present Robinson--Schensted--Knuth-type correspondences (that is, bijective correspondences between word arrays and certain pairs of semistandard tableaux of the same shape), generalizing two known correspondences: a bijective correspondence between standard words and certain pairs of standard tableaux, and an injective correspondence between words…

量子奇异超代数的晶体局部刻画

A local characterization of crystals for the quantum queer superalgebra

本文在半标准移位表上定义算子证明其晶体结构，并定义奇异晶体算子构造连通奇异晶体，推广 Stembridge 公理给出局部刻画，为 Schur $P$-多项式及其乘积的正性提供了新证明。

Determinantal and Pfaffian identities for ninth variation skew Schur functions and Q-functions

Recently Okada defined algebraically ninth variation skew Q-functions, in parallel to Macdonald's ninth variation skew Schur functions. Here we introduce a skew shifted tableaux definition of these ninth variation skew Q-functions, and prove by means of a non-intersecting lattice path model a Pfaffian outside decomposition result in the form of a ninth variation version of Hamel's Pfaffian outside decomposition identity. As corollaries to this we derive Pfaffian identities generalizing those of Josefiak-Pragacz, Nimmo, and most recently Okada. As a preamble to this we present a parallel develo…

Cyclic Sieving and Plethysm Coefficients

A combinatorial expression for the coefficient of the Schur function $s_λ$ in the expansion of the plethysm $p_{n/d}^d \circ s_μ$ is given for all $d$ dividing $n$ for the cases in which $n=2$ or $λ$ is rectangular. In these cases, the coefficient $\langle p_{n/d}^d \circ s_μ, s_λ \rangle$ is shown to count, up to sign, the number of fixed points of an $\langle s_μ^n, s_λ \rangle$-element set under the $d^{\text{th}}$ power of an order-$n$ cyclic action. If $n=2$, the action is the Schützenberger involution on semistandard Young tableaux (also known as evacuation), and, if $λ$ is rectangular, …

表概与表形代数的结构

Structure and geometry of the tableaux algebra

本文研究了半标准年轻表代数的结构与几何性质，证明其具有交换、Noetherian等代数性质，并分类了最大理想和不可约模，扩展了$\mathfrak{sl}_n$-crystals的嵌入结果。

Symplectic keys and Demazure atoms in type C

We compute, mimicking the Lascoux-Schützenberger type A combinatorial procedure, left and right keys for a Kashiwara-Nakashima tableau in type C. These symplectic keys have a similar role as the keys for semistandard Young tableaux. More precisely, our symplectic keys give a tableau criterion for the Bruhat order on the hyperoctahedral group and cosets, and describe Demazure atoms and characters in type C. The right and the left symplectic keys are related through the Lusztig involution. A type C Schützenberger evacuation is defined to realize that involution.

Permutation Tableaux and Permutation Patterns

In this paper we introduce and study a class of tableaux which we call permutation tableaux; these tableaux are naturally in bijection with permutations, and they are a distinguished subset of the Le-diagrams of Alex Postnikov. The structure of these tableaux is in some ways more transparent than the structure of permutations; therefore we believe that permutation tableaux will be useful in furthering the understanding of permutations. We give two bijections from permutation tableaux to permutations. The first bijection carries tableaux statistics to permutation statistics based on relative si…

A Ratio of Alternants Formula for Loop Schur Functions

Lam and Pylyavskyy introduced loop symmetric functions as a generalization of symmetric functions. They defined loop Schur functions as generating functions over semistandard tableaux with respect to a `colored weight,' and they proved a Jacobi--Trudi-style determinantal formula for these generating functions. We prove that loop Schur functions can be expressed as a ratio of `loop alternants,' extending the analogy with Schur functions. As an application, we give a new proof of the loop version of the Murnaghan--Nakayama rule.

通过围道积分与顶点模型推导斜形状的钩长公式

Hook-length Formulas for Skew Shapes via Contour Integrals and Vertex Models

本文通过多重围道积分和六顶点模型两种新方法，证明了斜形状标准杨表数量公式的多元推广，并将其扩展至标准杨表与 Schur 对称多项式之外。

The involutive nature of the Littlewood-Richardson commutativity bijection

Littlewood-Richardson (LR) coefficients $c_{μν}^λ$ may be evaluated by means of several combinatorial models, including the original LR tableaux of skew shape $λ/μ$ and weight $ν$ and the LR hives with boundary edge labels $λ$, $μ$ and $ν$. Unfortunately, neither of these reveal in any obvious way the well-known symmetry property $c_{μν}^λ=c_{νμ}^λ$. Here we introduce two maps, $ρ^{(n)}$ on LR tableaux and $σ^{(n)}$ on LR hives, that each interchange contributions to $c_{μν}^λ$ and $c_{νμ}^λ$ for any partitions $λ$, $μ$, $ν$ of lengths no greater than $n$, and then prove not only that each of …

链序多面体：环面退化、杨表与单项式基

Chain-order polytopes: toric degenerations, Young tableaux and monomial bases

本文将Gelfand--Tsetlin偏序集标记链序多面体的环面簇实现为旗簇的Gröbner退化，并利用pipe dreams将标准单项式理论与PBW-单项式基推广至任意MCOP，同时引入半无穷pipe dreams构造半无穷Grassmannian的环面退化族。

关于 barely set-valued 表列与平面分拆的 $q$-计数

On the $q$-Enumeration of Barely Set-Valued Tableaux and Plane Partitions

该研究针对 barely set-valued 表列与平面分拆的计数问题，引入主指标版本给出相关乘积公式的$q$-模拟，以$P$-分拆一般形式表述结果并特化得到简洁公式，所用偏序集序理想上的概率分布也具独立价值。