Proof of a partition identity conjectured by Lassalle

A Multiple Integral Explicit Evaluation Inspired by the Multi-WZ Method

We give an identity which is conjectured and proved by using an implementation in Multi-WZ.

Franklin恒等式的一个周长类比及与部分奇偶性相关的不等式

A perimeter analogue of Franklin's identity and an inequality related to the parity of parts

本文通过生成函数与渐近分析，证明了关于分拆周长不等式的两个猜想，并建立了与无峰Motzkin路径的联系。

A Unified Combinatorial Approach to Several Poincare Series Identities

Mendes recently conjectured an identity simplifying the Poincaré series of the space of equivariant polynomial maps from $\mathbb{R}^{n}$ to a subrepresentation of $Sym^{2}(\mathbb{R}^{n})$. We show how to prove this identity using a fairly simple integer partition bijection. First, we give a bijective proof of a similar, well-known identity from representation theory. We then show that this bijection can be generalized to prove other Poincaré series identities, including a version of the identity conjectured by Mendes as well as refinements of it.

Proofs of some partition identities conjectured by Kanade and Russell

Kanade and Russell conjectured several Rogers-Ramanujan-type partition identities, some of which are related to level $2$ characters of the affine Lie algebra $A_9^{(2)}$. Many of these conjectures have been proved by Bringmann, Jennings-Shaffer and Mahlburg. We give new proofs of five conjectures first proved by those authors, as well as four others that have been open until now. Our proofs for the new cases use quadratic transformations for Askey-Wilson and Rogers polynomials. We also obtain some related results, including a new proof of a partition identity conjectured by Capparelli and fir…

A proof of the McKay-Radziszowski subgraph counting conjecture

We prove a theorem on how to count induced subgraphs in neighborhoods of graphs. Then we use it to prove a subgraph counting identity conjectured by McKay and Radziszowski in there work on Ramsey theory.

A uniform proof of the Macdonald-Mehta-Opdam identity for finite Coxeter groups

We give a new proof of the Macdonald-Mehta-Opdam integral identity for finite Coxeter groups. This identity was conjectured by Macdonald and proved by Opdam in 1993 using the theory of multivariable Bessel functions, but in non-crystallographic cases the proof relied on a computer calculation by F. Garvan. Our proof is somewhat more elementary (in particular, it does not use multivariable Bessel functions), and uniform (does not refer to the classification of finite Coxeter groups and does not use computers).

Une identité remarquable en théorie des partitions

We prove an identity about partitions, previously conjectured in the study of shifted Jack polynomials (math.CO/9903020). The proof given is using $λ$-ring techniques. It would be interesting to obtain a bijective proof.

Descents of $λ$-unimodal cycles in a character formula

We prove an identity conjectured by Adin and Roichman involving the descent set of $λ$-unimodal cyclic permutations. These permutations appear in formulas for characters of certain representations of the symmetric group. Such formulas have previously been proven algebraically. In this paper, we present a combinatorial proof for one such formula and discuss the consequences for the distribution of the descent set on cyclic permutations.

Demonstration de la conjecture de Dumont

Dumont has conjectured a marvellous identity, which generalizes, in particular, the classical results of Lagrange, Gauss, Jacobi and Kronecker on the sums of two, three and four squares. We give a combinatorial proof of Dumont's conjecture.

一个将Catalan数与tangent数联系起来并具有算术应用的恒等式

An identity relating Catalan numbers to tangent numbers with arithmetic applications

本文证明了一个将Catalan数与tangent数联系起来的组合恒等式，由此引出新的恒等式并应用于证明整除性质，同时给出了其$q$-模拟及组合证明。

Generalized Alder-Type Partition Inequalities

In 2020, Kang and Park conjectured a "level $2$" Alder-type partition inequality which encompasses the second Rogers-Ramanujan Identity. Duncan, Khunger, the fourth author, and Tamura proved Kang and Park's conjecture for all but finitely many cases utilizing a "shift" inequality and conjectured a further, weaker generalization that would extend both Alder's (now proven) as well as Kang and Park's conjecture to general level. Utilizing a modified shift inequality, Inagaki and Tamura have recently proven that the Kang and Park conjecture holds for level $3$ in all but finitely many cases. They …

On a Partition Identity of Lehmer

Euler's identity equates the number of partitions of any non-negative integer n into odd parts and the number of partitions of n into distinct parts. Beck conjectured and Andrews proved the following companion to Euler's identity: the excess of the number of parts in all partitions of n into odd parts over the number of parts in all partitions of n into distinct parts equals the number of partitions of n with exactly one even part (possibly repeated). Beck's original conjecture was followed by generalizations and so-called "Beck-type" companions to other identities. In this paper, we establish…

The Pelletier-Ressayre hidden symmetry for Littlewood-Richardson coefficients

We prove an identity for Littlewood--Richardson coefficients conjectured by Pelletier and Ressayre (arXiv:2005.09877). The proof relies on a novel birational involution defined over any semifield.

多分量$q$-Baker–Forrester猜想的一个证明

A proof of the multi-component $q$-Baker--Forrester conjecture

本文证明了1998年由Baker与Forrester提出的$(p+1)$分量$q$-Baker–Forrester猜想，解决了这一悬置26年的数学猜想，该猜想是$q$-Morris恒等式的多分量推广，可推导出Selberg积分的推广形式。

关于类型 $D_k$ 卡坦矩阵的纳姆求和恒等式

Nahm sum identities for Cartan matrices of type $D_k$

本文通过贝利配对理论重新证明了与类型 $D_k$ 卡坦矩阵相关的四个纳姆求和恒等式，并给出了其参数推广的两种不同证明。

多重zeta值的组合方面

Combinatorial aspects of multiple zeta values

本文利用多重zeta值的积分表示满足洗牌乘积规则，采用组合方法证明了Zagier猜想及类似循环和恒等式，并提出支持无限族推广恒等式的计算证据。

Off-critical parafermions and the winding angle distribution of the O($n$) model

Using an off-critical deformation of the identity of Duminil-Copin and Smirnov, we prove a relationship between half-plane surface critical exponents $γ_1$ and $γ_{11}$ as well as wedge critical exponents $γ_2(α)$ and $γ_{21}(α)$ and the exponent characterising the winding angle distribution of the O($n$) model in the half-plane, or more generally in a wedge of wedge-angle $α.$ We assume only the existence of these exponents and, for some values of $n,$ the conjectured value of the critical point. If we assume their values as predicted by conformal field theory, one gets complete agreement wit…

致力于解决 Kang 和 Park 对 Alder-Andrews 定理的推广

Toward resolving Kang and Park's generalization of the Alder-Andrews Theorem

本文通过推广 Alfes 等人的渐近方法与高性能计算技术，解决了 Kang 和 Park 提出的与第二个 Rogers-Ramanujan 恒等式相关的“2 级”Alder-Andrews 型分拆不等式猜想中几乎所有剩余情况。

Prellberg–Stanton非负性猜想的证明

Proof of a nonnegativity conjecture of Prellberg--Stanton

本文通过证明另一幂级数的系数逐点非负这一更强结果，推导出Prellberg–Stanton猜想，证明用到高斯多项式的莫比乌斯修正、分拆的保权单射等方法。

偶模数下 Bressoud 猜想的 overpartition 类比

An overpartition analogue of Bressoud conjecture for even moduli

本文通过引入新的分拆函数 $\overline{B}_0$ 并构造其与 $B_1$ 之间的双射，证明了 $j=0$ 情形下 Bressoud 猜想的 overpartition 类比。

Gromov-Witten理论中两个多元$q$-二项式和的证明

Proof of Two Multivariate $q$-Binomial Sums Arising in Gromov-Witten Theory

本研究证明了Gromov-Witten理论中两个与特定对数Calabi-Yau曲面不变量生成级数相关的多元$q$-二项式猜想恒等式，核心工具是基本超几何级数中的Jackson的Pfaff-Saalschütz求和公式的$q$-模拟。

关于固定分拆周长的一个猜想的证明

A Proof of a Conjecture on Fixed Perimeter Partitions

本文证实了Gray等人提出的对所有n≥9成立的奇偶偏差不等式的固定周长类似猜想，该猜想源于分拆理论的固定周长相关研究。

森德拉姆有界区间高阶李正性猜想的一个证明

A Proof of Sundaram's Bounded-Interval Higher Lie Positivity Conjecture

证明森德拉姆关于\(F_{n,M}\)舒尔展开式系数非负整数的猜想，通过幂和展开、伯恩斯坦公式等方法，利用杨图划分、特征值分离等技术，还给出相关反例。

树计数中的锐度准则与Kuperberg的G₂蜘蛛中的三角剖分渐近数量

A criterion for sharpness in tree enumeration and the asymptotic number of triangulations in Kuperberg's G2 spider

本文证明了Kuperberg关于李代数$G_2$表示论的三角剖分渐近公式猜想，提出树计数锐度准则并通过奇点分析细化了正则$n$边形三角剖分数量的渐近估计。

关于巴拉的广义m表示的调和数证明

A Proof of Bala's General-$m$ Representation of the Harmonic Numbers

本文证明了巴拉提出的广义m表示调和数的恒等式，通过形式幂级数方法推导出该恒等式成立，适用于任意非零整数m。