Emmanuel Briand
We give three proofs of the following result conjectured by Carriegos, De Castro-Garc\'ıa and Muñoz Castañeda in their work on enumeration of control systems: when $\binom{k+1}{2} \le n < \binom{k+2}{2}$, there are as many partitions of $n$ with $k$ corners as pairs of partitions $(α, β)$ such that $\binom{k+1}{2} + |α| + |β| = n$.
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We give a product formula for the number of shifted plane partitions of shifted double staircase shape with bounded entries. This is the first new example of a family of shapes with a plane partition product formula in many years. The proof is based on the theory of lozenge tilings; specifically, we apply the "free boundary" Kuo condensation due to Ciucu.
Cyclic Sieving for Staircase Plane Partitions via Crystals and Electrical Networks
研究高度为二的阶梯平面划分提升作用的循环筛选结果，利用自旋晶体张量幂解释及电网络空间坐标环二次部分的灌木丛基证明，还探讨了所有度数上电典范基存在时对所有高度阶梯平面划分提升的循环筛选意义。
In their 2002 paper, Ciucu and Krattenthaler proved several product formulas for the number of lozenge tilings of various regions obtained from a centrally symmetric hexagon on the triangular lattice by removing maximal staircase regions from two non-adjacent corners. For the case when the staircases are removed from adjacent corners of the hexagon, they presented two conjectural formulas, whose proofs, as they remarked, seemed at the time "a formidable task". In this paper we prove those two conjectures. Our proofs proceed by first generalizing the conjectures, and then proving them by induct…
Macdonald principal specialization ratios and their generating polynomials
本文研究广义阶梯划分，通过有限主特化公式推导连续阶梯间的最短序列比值，证明非负性条件，并给出乘积公式及特化下的应用。
(Dual-)promotion and (dual-)evacuation are bijections on SYT(λ) for any partition λ. Let c^r denote the rectangular partition (c,...,c) of height r, and let sc_k (k > 2) denote the staircase partition (k,k-1,...,1). B. Rhoades showed representation-theoretically that promotion on SYT(c^r) exhibits the cyclic sieving phenomenon (CSP). In this paper, we demonstrate a promotion- and evacuation-preserving embedding of SYT(sc_k) into SYT(k^{k+1}). This arose from an attempt to demonstrate the CSP of promotion action on SYT(sc_k).
A bijection between symmetric plane partitions and quasi transpose complementary plane partitions
本文提出一种双射，将对称平面划分与准转置互补平面划分联系起来，并解决了一个35年的开放问题。
Staircase hook-length ratios and special values of Jacobi polynomials
本文通过将阶梯划分的钩长乘积与雅可比多项式特殊值联系起来，推导出稳定格罗滕迪克多项式和 $K$-理论 Schur $P$-函数的特殊值，并揭示了集合值半标准 Young 表数目的比值。
We use intermediate symplectic characters to give a proof and variations of Hopkins' conjecture, now proved by Hopkins and Lai, on the number of shifted plane partitions of shifted double staircase shape with bounded entries. In fact, we prove some character identities involving intermediate symplectic characters, and find generating functions for such shifted plane partitions. The key ingredients of the proof are a bialternant formula for intermediate symplectic characters, which interpolates between those for Schur functions and symplectic characters, and the Ishikawa-Wakayama minor-summatio…
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…
We utilize the technique of staircases and jagged partitions to provide analytic sum-sides to some old and new partition identities of Rogers-Ramanujan type. Firstly, we conjecture a class of new partition identities related to the principally specialized characters of certain level $2$ modules for the affine Lie algebra $A_9^{(2)}$. Secondly, we provide analytic sum-sides to some earlier conjectures of the authors. Next, we use these analytic sum-sides to discover a number of further generalizations. Lastly, we apply this technique to the well-known Capparelli identities and present analytic …
Staircase Minimality and a Proof of Saxl's Conjecture
本文证明了阶梯分区的张量平方包含所有不可约表示，同时证明了阶梯是唯一Kronecker普遍自共轭分区，完善了Saxl猜想的结论。
Motivated by the enumeration of a class of plane partitions studied by Proctor and by considerations about symmetry classes of plane partitions, we consider the problem of enumerating lozenge tilings of a hexagon with ``maximal staircases'' removed from some of its vertices. The case of one vertex corresponds to Proctor's problem. For two vertices there are several cases to consider, and most of them lead to nice enumeration formulas. For three or more vertices there do not seem to exist nice product formulas in general, but in one special situation a lot of factorization occurs, and we pose t…
In 2012 Jan Saxl conjectured that all irreducible representations of the symmetric group occur in the decomposition of the tensor square of the irreducible representation corresponding to the staircase partition. We make progress on this conjecture by proving the occurrence of all those irreducibles which correspond to partitions that are comparable to the staircase partition in the dominance order. Moreover, we use our result to show the occurrence of all irreducibles corresponding to hook partitions. This generalizes results by Pak, Panova, and Vallejo from 2013.
We explore some probabilistic applications arising in connections with $K$-theoretic symmetric functions. For instance, we determine certain corner distributions of random lozenge tilings and plane partitions. We also introduce some distributions that are naturally related to the corner growth model. Our main tools are dual symmetric Grothendieck polynomials and normalized Schur functions.
On equidistribution theorem for plane partitions
本文证明了平面分割上两个统计量对的等分布性，并引入了新的统计量证明其等分布性。
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…
The "n! conjecture" of Garsia and Haiman has inspired mathematicians for nearly two decades, even after Haiman published a proof in 2001. Kumar and Funch Thomsen proved in 2003 that in order to prove the conjecture for all partitions, it suffices to prove it for the so-called "staircase partitions" $(k,k-1,...,2,1)$ for each $k>1$. In the present paper we give a construction of a specially designed two-dimensional family of length-$n$ subschemes of the plane, and use that to prove the $n!$ conjecture for staircase partitions. Together with the result of Kumar and Funch Thomsen, this provides a…
Refined Enumeration of Corners in Tree-like Tableaux
本文通过中间结构确认了树状表格及对称树状表格中非占用角落精细枚举的两个猜想，涉及替代表格和链接划分等工具。
Ehrhart reciprocity and forced factors in three plane-partition enumerators
本文研究三个平面分拆计数函数，利用 Ehrhart-Macdonald 互反性实现因式分解，确定线性因子、剩余多项式性质及分母整除界，并验证小尺寸下的不可约性。
Following the breakthrough of Croot, Lev, and Pach, Tao introduced a symmetrized version of their argument, which is now known as the slice rank method. In this paper, we introduce a more general version of the slice rank of a tensor, which we call the Partition Rank. This allows us to extend the slice rank method to problems that require the variables to be distinct. Using the partition rank, we generalize a recent result of Ge and Shangguan, and prove that any set $A\subset\mathbb{F}_{q}^{n}$ of size \[|A|>\binom{n+(k-1)q}{(k-1)(q-1)}\] contains a $k$-right-corner, that is distinct vectors $…
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 $|{…
Symmetry of ascent and descent distributions on rectangular and staircase tableaux
本文利用标准 Young 表的箭头编码构造双射，证明矩形与截断阶梯形表上上升、下降分布对称，并推广到统计量保持对合、Sulanke 猜想、rowmotion 及 canon 排列。
Triangular partitions: enumeration, structure, and generation
本文研究了三角划分，给出了其新刻画、Möbius 函数公式、适合正方形的计数公式，并提出了高效生成算法，可计算大小至 10^5 的三角划分数量。
Hecke growth diagrams, and maximal increasing and decreasing sequences in fillings of stack polyominoes
本文通过建立堆叠多联骨牌$01$-填充与拐角标记的双射，推广了增长图技术，为最长递增递减链对称分布定理提供了新证明与视角。
We introduce the structure of vacillating Hecke tableaux, and establish a one-to-one correspondence between vacillating Hecke tableaux and linked partitions by using the Hecke insertion algorithm developed by Buch, Kresch, Shimozono, Tamvakis and Yong. Linked partitions arise in free probability theory. Motivated by the Hecke insertion algorithm, we define a Hecke diagram as a Young diagram possibly with a marked corner. A vacillating Hecke tableau is defined as a sequence of Hecke diagrams subject to certain addition and deletion of rook strips. The notion of a rook strip was introduced by Bu…
Coloured corner processes from asymptotics of LLT polynomials
研究秩 $n \geq 1$ 的 LLT 对称多项式 Cauchy 恒等式导出的概率测度，利用费米子格点模型和围道积分表示，证明其在 $n>1$ 时渐近分裂为 $n$ 个 GUE 角过程乘积的连续部分与含交错着色的离散部分。
Tverberg's theorem is one of the cornerstones of discrete geometry. It states that, given a set $X$ of at least $(d+1)(r-1)+1$ points in $\mathbb R^d$, one can find a partition $X=X_1\cup \ldots \cup X_r$ of $X$, such that the convex hulls of the $X_i$, $i=1,\ldots,r$, all share a common point. In this paper, we prove a strengthening of this theorem that guarantees a partition which, in addition to the above, has the property that the boundaries of full-dimensional convex hulls have pairwise nonempty intersections. Possible generalizations and algorithmic aspects are also discussed. As a concr…
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