Andrei Okounkov, Nicolai Reshetikhin
We study random skew 3D partitions weighted by $q^{\textup{vol}}$ and, specifically, the $q\to 1$ asymptotics of local correlations near various points of the limit shape. We obtain sine-kernel asymptotics for correlations in the bulk of the disordered region, Airy kernel asymptotics near a general point of the frozen boundary, and a Pearcey kernel asymptotics near a cusp of the frozen boundary.
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Universal Cusp Scaling in Random Partitions
本文研究Schur测度下随机划分的普适标度极限，扩展先前工作获得高阶Pearcey核，并推导间隙概率的耦合非线性微分方程及大间隙渐近行为。
Random skew plane partitions of large size distributed according to an appropriately scaled Schur process develop limit shapes. In the present work we consider the limit of large random skew plane partitions where the inner boundary approaches a piecewise linear curve with non-lattice slopes, describing the limit shape and the local fluctuations in various regions. This analysis is fairly similar to that in [OR2], but we do find some new behavior. For instance, the boundary of the limit shape is now a single smooth (not algebraic) curve, whereas the boundary in [OR2] is singular. We also obser…
Inspired by the works of Dewar, Murty and Kotěšovec, we establish some useful theorems for asymptotic formulas. As an application, we obtain asymptotic formulas for the numbers of skew plane partitions and cylindric partitions. We prove that the order of the asymptotic formula for the skew plane partitions of fixed width depends only on the width of the region, not on the profile (the skew zone) itself, while this is not true for cylindric partitions.
Limit shapes and fluctuations for $(GL_n, GL_k)$ skew Howe duality
本文研究双柯西恒等式中Schur多项式特化诱导的Young图概率测度，证明极限形状存在，并刻画体相与边界涨落分别由离散正弦核和Tracy-Widom分布等控制，恢复已知离散分布并发现Pearcey核出现条件。
In this paper we give an easy combinatorial description for the base partition B of a skew character [A], which is the intersection of all partitions alpha whose corresponding character [alpha] appears in [A]. This we use to construct the cover partition C for the ordinary outer product as well as for the Schubert product of two characters and for some skew characters, here the cover partition is the union of all partitions whose corresponding character appears in the product or in the skew character. This gives us also the Durfee size for arbitrary Schubert products.
In this paper we construct for a given arbitrary skew diagram A all partitions nu with maximal principal hook lengths among all partitions with the character [nu] appearing in the skew character [A]. Furthermore we show that these are also partitions with minimal Durfee size. This we use to give the maximal Durfee size for [nu] appearing in [A] for the cases when A decays into two partitions and for some special cases of A. Also this gives conditions for two skew diagrams to represent the same skew character.
Let the sign of a skew standard Young tableau be the sign of the permutation you get by reading it row by row from left to right, like a book. We examine how the sign property is transferred by the skew Robinson-Schensted correspondence invented by Sagan and Stanley. The result is a remarkably simple generalization of the ordinary non-skew formula. The sum of the signs of all standard tableaux on a given skew shape is the sign-imbalance of that shape. We generalize previous results on the sign-imbalance of ordinary partition shapes to skew ones.
Even pairs in Berge graphs with no balanced skew-partitions
本文研究无奇棱镜和长反洞诱导子图的Berge图，证明在无平衡斜划分条件下，这类图要么是完全图，要么必含偶对。
Nazarov and Tarasov recently generalized the notion of the rank of a partition to skew partitions. We give several characterizations of the rank of a skew partition and one possible characterization that remains open. One of the characterizations involves the decomposition of a skew shape into a minimal number of border strips, and we develop a theory of these MBSD's as well as of the closely related minimal border strip tableaux. An application is given to the value of a character of the symmetric group S_n indexed by a skew shape z at a permutation whose number of cycles is the rank of z.
We present an affirmative answer to Stanley's zrank problem, namely, the zrank and rank are equal for any skew partition. We show that certain classes of restricted Cauchy matrices are nonsingular and furthermore, the signs depend on the number of zero entries. Similar to notion of the jrank of a skew partition, we give a characterization of the rank in terms of the Giambelli type matrices of the corresponding skew Schur functions. We also show that the sign of the determinant of a factorial Cauchy matrix is uniquely determined by the number of its zero entries, which implies the nonsingularit…
The paper studies scaling limits of random skew plane partitions confined to a box when the inner shapes converge uniformly to a piecewise linear function V of arbitrary slopes in [-1,1]. It is shown that the correlation kernels in the bulk are given by the incomplete Beta kernel, as expected. As a consequence it is established that the local correlation functions in the scaling limit do not depend on the particular sequence of discrete inner shapes that converge to V. A detailed analysis of the correlation kernels at the top of the limit shape and of the frozen boundary is given. It is shown …
In this paper, which is a complement of \cite{BG}, we study a few elementary invariants for configurations of skew lines, as introduced and analyzed first by Viro and his collaborators. We slightly simplify the exposition of some known invariants and use them to define a natural partition of the lines in a skew configuration. We also describe an algorithm which constructs a spindle-permutation for a given switching class, or proves non-existence of such a spindle-permutation.
A hole in a graph is an induced cycle on at least four vertices. A graph is Berge if it has no odd hole and if its complement has no odd hole. In 2002, Chudnovsky, Robertson, Seymour and Thomas proved a decomposition theorem for Berge graphs saying that every Berge graph either is in a well understood basic class, or has some kind of decomposition. Then, Chudnovsky proved stronger theorems. One of them restricts the allowed decompositions to 2-joins and balanced skew partitions. We prove that the problem of deciding whether a graph has a balanced skew partition is NP-hard. We give an $O(n^9)$-…
Plane partitions have been widely studied in Mathematics since MacMahon. See, for example, the works by Andrews, Macdonald, Stanley, Sagan and Krattenthaler. The Schur process approach, introduced by Okounkov and Reshetikhin, and further developed by Borodin, Corwin, Corteel, Savelief and Vuletić, has been proved to be a powerful tool in the study of various kinds of plane partitions. The exact enumerations of ordinary plane partitions, shifted plane partitions and cylindric partitions could be derived from two summation formulas for Schur processes, namely, the open summation formula and the …
We study edge asymptotics of poissonized Plancherel-type measures on skew Young diagrams (integer partitions). These measures can be seen as generalizations of those studied by Baik--Deift--Johansson and Baik--Rains in resolving Ulam's problem on longest increasing subsequences of random permutations and the last passage percolation (corner growth) discrete versions thereof. Moreover they interpolate between said measures and the uniform measure on partitions. In the new KPZ-like 1/3 exponent edge scaling limit with logarithmic corrections, we find new probability distributions generalizing th…
Free skew Boolean intersection algebras and set partitions
本文建立自由左斜布尔交代数原子与集合划分的双射，描述有限结构并用贝尔数和斯特林数表达组合特征，同时推广到无限情形。
The Pak--Postnikov and Naruse skew hook length formulas: a new proof
本文将 Pak–Postnikov 的生成函数推广与 Naruse 的斜分拆推广结合为斜分拆的生成函数恒等式，并用递归、行列式和简单组合方法给出一个较为初等的新证明。
It is known that the Schur expansion of a skew Schur function runs over the interval of partitions, equipped with dominance order, defined by the least and the most dominant Littlewood-Richardson filling of the skew shape. We characterise skew Schur functions (and therefore the product of two Schur functions) which are multiplicity-free and the resulting Schur expansion runs over the whole interval of partitions, i.e. skew Schur functions having Littlewood-Richardson coefficients always equal to $1$ over the full interval.
Cyclotomic skew partial difference sets and partial difference families
本文研究了循环理论在构造斜部分差集中的应用，提出了新的构造方法，并探讨了斜部分差集与部分差族之间的关系。
Recently, Naruse discovered a hook length formula for the number of standard Young tableaux of a skew shape. Morales, Pak and Panova found two $q$-analogs of Naruse's hook length formula over semistandard Young tableaux (SSYTs) and reverse plane partitions (RPPs). As an application of their formula, they expressed certain $q$-Euler numbers, which are generating functions for SSYTs and RPPs of a zigzag border strip, in terms of weighted Dyck paths. They found a determinantal formula for the generating function for SSYTs of a skew staircase shape and proposed two conjectures related to RPPs of t…
Peterson and Proctor obtained a formula which expresses the multivariate generating function for $P$-partitions on a $d$-complete poset $P$ as a product in terms of hooks in $P$. In this paper, we give a skew generalization of Peterson--Proctor's hook formula, i.e., a formula for the generating function for $(P \setminus F)$-partitions for a $d$-complete poset $P$ and its order filter $F$, by using the notion of excited diagrams. Our proof uses the Billey-type formula and the Chevalley-type formula in the equivariant $K$-theory of Kac--Moody partial flag varieties. This generalization provides…
Inspired by a question of Yannakakis on the Vertex Packing polytope of perfect graphs, we study the Clique-Stable Set Separation in a non-hereditary subclass of perfect graphs. A cut (B,W) of G (a bipartition of V(G)) separates a clique K and a stable set S if $K\subseteq B$ and $S\subseteq W$. A Clique-Stable Set Separator is a family of cuts such that for every clique K, and for every stable set S disjoint from K, there exists a cut in the family that separates K and S. Given a class of graphs, the question is to know whether every graph of the class admits a Clique-Stable Set Separator cont…
We show that the shifted rank, or srank, of any partition $λ$ with distinct parts equals the lowest degree of the terms appearing in the expansion of Schur's $Q_λ$ function in terms of power sum symmetric functions. This gives an affirmative answer to a conjecture of Clifford. As pointed out by Clifford, the notion of the srank can be naturally extended to a skew partition $λ/μ$ as the minimum number of bars among the corresponding skew bar tableaux. While the srank conjecture is not valid for skew partitions, we give an algorithm to compute the srank.
This article presents uniform random generators of plane partitions according to the size (the number of cubes in the 3D interpretation). Combining a bijection of Pak with the method of Boltzmann sampling, we obtain random samplers that are slightly superlinear: the complexity is $O(n (\ln n)^3)$ in approximate-size sampling and $O(n^{4/3})$ in exact-size sampling (under a real-arithmetic computation model). To our knowledge, these are the first polynomial-time samplers for plane partitions according to the size (there exist polynomial-time samplers of another type, which draw plane partitions…
A non-iterative straightening algorithm and orthogonality for skew Schur modules
本文将 Fulton 的 Schur 模行列式构造推广至斜设置，提出了一种非迭代拉直算法以显式闭系数公式将任意元素表示为新的 D-基，并证明该基源于对半标准表基的 Gram-Schmidt 正交化，从而揭示了自然内积并将拉直重表述为正交投影。
We prove and generalize a conjecture in arXiv:1610.0474(4) about the asymptotics of $\frac{1}{\sqrt{n!}} f^{λ/μ}$, where $f^{λ/μ}$ is the number of standard Young tableaux of skew shape $λ/μ$ which have stable limit shape under the $1/\sqrt{n}$ scaling. The proof is based on the variational principle on the partition function of certain weighted lozenge tilings.
We study large random partitions boxed into a rectangle and coming from skew Howe duality, or alternatively from dual Schur measures. As the sides of the rectangle go to infinity, we obtain: 1) limit shape results for the profiles generalizing the Vershik--Kerov--Logan--Shepp curve; and 2) universal edge asymptotic results for the first parts in the form of the Tracy--Widom distribution, as well as less-universal critical regime results introduced by Gravner, Tracy and Widom. We do this for a large class of Schur parameters going beyond the Plancherel or principal specializations previously st…
The Symplectic Schur Process
本文定义了辛施尔过程，该过程是Okounkov-Reshetikhin对于Cartan型C的类比。通过引入辛字符和新的Down-Up施尔函数，证明了相关点过程的行列性，并给出了显式相关核。还研究了保形过程的动力学和采样方案，以及极限行为。
For each skew shape we define a nonhomogeneous symmetric function, generalizing a construction of Pak and Postnikov. In two special cases, we show that the coefficients of this function when expanded in the complete homogeneous basis are given in terms of the (reduced) type of $k$-divisible noncrossing partitions. Our work extends Haiman's notion of a parking function symmetric function.
Cylindric skew Schur functions, which are a generalisation of skew Schur functions, arise naturally in the study of P-partitions. Also, recent work of A. Postnikov shows they have a strong connection with a problem of considerable current interest: that of finding a combinatorial proof of the non-negativity of the 3-point Gromov-Witten invariants. After explaining these motivations, we study cylindric skew Schur functions from the point of view of Schur-positivity. Using a result of I. Gessel and C. Krattenthaler, we generalise a formula of A. Bertram, I. Ciocan-Fontanine and W. Fulton, thus g…
Positivity among P-partition generating functions
研究标号偏序集对$(P,ω)$-分拆枚举数之差为$F$-正的条件，作为斜Schur函数正性问题的拟对称类比，给出了必要、充分及充要条件并证明组合运算保持正性。
We study scaling limits of skew plane partitions with periodic weights under several boundary conditions. We compute the correlation kernel of the limiting point process in the bulk and near turning points on the frozen boundary. The turning points that appear in the homogeneous case split in our model into pairs of turning points macroscopically separated by a "semi-frozen" region. As a result the point process at a turning point is not the GUE minor process, but rather a pair of GUE minor processes, non-trivially correlated. We also study an intermediate regime when the weights are periodic …
Regular Schur labeled skew shape posets and their 0-Hecke modules
该研究在Stanley的$P$-分拆猜想假设下，刻画了正则Schur标记斜形状偏序集的性质，分类其线性扩张集合与相关0-Hecke模，并构造了模关于Schur基的特异滤过。
Cyclic sieving, skew Macdonald polynomials and Schur positivity
该研究证明了分拆部分为n的倍数时，特殊化非对称麦克唐纳多项式对应的填充集合满足循环筛分现象，引入其斜版本并证明Schur正性，还得到一类新LLT多项式的组合Schur展开。
On $P$-partitions Extended by Two-Rowed Plane Partitions
本文引入 $φ_{z}$、$ψ_{z}$ 两个算子，建立两行平面分拆扩展偏序集的 $P$-分拆新计数公式，并推广到多种平面分拆与偏序集结构。
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…
On the combinatorics of the refined 1-leg DT/PT correspondence
本文通过几何DT/PT墙穿越公式重新证明了Bessenrodt的结果，并建立了反平面分区和斜平面分区的加权计数新公式，同时引入了Young图内部和外部钩的生成函数新恒等式。
Generalized Pitman-Stanley polytope: vertices and faces
本文研究 Pitman-Stanley 多胞体的推广，证明其可实现为网格图的流多胞体，并给出顶点刻画、顶点数多项式及面数与生成函数公式。
For non-negative integers $n$ and $k$ with $n \ge k$, a {\em $k$-minor} of a partition $λ= [λ_1, λ_2, \dots]$ of $n$ is a partition $μ= [μ_1, μ_2, \dots]$ of $n-k$ such that $μ_i \le λ_i$ for all $i$. The multiset $\widehat{M}_k(λ)$ of $k$-minors of $λ$ is defined as the multiset of $k$-minors $μ$ with multiplicity of $μ$ equal to the number of standard Young tableaux of skew shape $λ/ μ$. We show that there exists a function $G(n)$ such that the partitions of $n$ can be reconstructed from their multisets of $k$-minors if and only if $k \le G(n)$. Furthermore, we prove that $\lim_{n \rightarro…
Demazure crystal structure for flagged reverse plane partitions
本文研究带标志反向平面分拆，证明其集合可分解为 Demazure 晶体的不交并，并据此推出带标志对偶稳定 Grothendieck 多项式具有 key 正性。
The geometry of the real four-qubit Pauli group, being embodied in the structure of the symplectic polar space W(7,2), is analyzed in terms of ovoids of a hyperbolic quadric of PG(7,2), the seven-dimensional projective space of order two. The quadric is selected in such a way that it contains all 135 symmetric elements of the group. Under such circumstances, the third element on the line defined by any two points of an ovoid is skew-symmetric, as is the nucleus of the conic defined by any three points of an ovoid. Each ovoid thus yields 36/84 elements of the former/latter type, accounting for …
Probability-theoretic interpretation of degeneracy locus formulas
本文用随机有理六顶点模型解释退化轨迹的Schwartz-MacPherson类，完成三种经典类型的稳定SSM理论，将斜类与对称类分别解释为概率和Chebyshev矩。
Asymptotics of parity biases for partitions into distinct parts via Nahm sums
本文推广了Kim等人的奇偶偏差研究，利用圆法给出互异分拆的模N偏差渐近结果，同时提供了分析Nahm型q-超几何级数渐近性的独立有用技术。
Demazure crystals for flagged set-valued reverse plane partitions
本文在带旗集值反向平面分割上构建Demazure晶体，统一推广了已知结构，并由此得到混合Grothendieck多项式的key多项式展开，进而将其表示为稳定与对偶稳定Grothendieck多项式的组合。
The rank of a skew partition $λ/μ$, denoted $rank(λ/μ)$, is the smallest number $r$ such that $λ/μ$ is a disjoint union of $r$ border strips. Let $s_{λ/μ}(1^t)$ denote the skew Schur function $s_{λ/μ}$ evaluated at $x_1=...=x_t=1, x_i=0$ for $i>t$. The zrank of $λ/μ$, denoted $zrank(λ/μ)$, is the exponent of the largest power of $t$ dividing $s_{λ/μ}(1^t)$. Stanley conjectured that $rank(λ/μ)=zrank(λ/μ)$. We show the equivalence between the validity of the zrank conjecture and the nonsingularity of restricted Cauchy matrices. In support of Stanley's conjecture we give affirmative answers for s…
Lower and Upper Bounds for Nonzero Littlewood-Richardson Coefficients
本文针对非零Littlewood-Richardson系数$c^γ_{λ,μ}>0$的条件，提出了一种基于Littlewood-Richardson表数量刻画和（广义）支配序的算法，以确定划分$μ$必须满足的下界和上界。
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