The q-Log-Concavity and Unimodality of q-Kaplansky Numbers

类型 $A_n$ 扇形坐标中簇单项式的对数凹性与单峰性

Log-concavity and unimodality of cluster monomials in fan coordinates of type $A_n$

本文证明类型 $A_n$ 的簇单项式在扇形坐标中具有对数凹性与单峰性，通过系数公式、整数线归约和卷积方法，部分证实了相关猜想。

Frobenius 极大抛物子代数的 Ooms 谱：严格单峰性与欧几里得对数凹性

Ooms spectra of Frobenius maximal parabolics: strict unimodality and Euclidean log-concavity

研究 Frobenius 极大抛物子代数的 Ooms 谱，证明严格单峰性并借助欧几里得算法和有限截断验证对数凹性，覆盖较小块至多 128 的情形。

曲面上点的Hilbert概型的Hodge数的对数凹性与单峰性

Log-concavity and unimodality of Hodge numbers of Hilbert schemes of points over a surface

本研究针对光滑复射影曲面的n点Hilbert概型，证明了其Hodge数序列的对数凹性等价于g≤组合数且由n=2的序列决定，单峰性等价于q≥1或g=0且由n=1的序列判定。

Unimodality on $δ$-vectors of lattice polytopes and two related properties

In this paper, we investigate two properties concerning the unimodality of the $δ$-vectors of lattice polytopes, which are log-concavity and alternatingly increasingness. For lattice polytopes $\mathcal{P}$ of dimension $d$, we prove that the dilated lattice polytopes $n\mathcal{P}$ have strictly log-concave and strictly alternatingly increasing $δ$-vectors if $n > \max\{s,d+1-s\}$, where $s$ is the degree of the $δ$-polynomial of $\mathcal{P}$. The bound $\max\{s,d+1-s\}$ for $n$ is reasonable. We also provide several kinds of unimodal (or non-unimodal) $δ$-vectors. Concretely, we give exampl…

Ehrhart 理论中的单峰性反例

Unimodality shenanigans in Ehrhart theory

本文通过构造矩形棱柱的 Cayley 和，否证了 Stanley、Brenti 及 Ferroni-Higashitani 关于 IDP 多胞体 $h^*$-多项式单峰性与对数凹性的多个猜想，并给出光滑反例。

拟阵平坦计数不是单峰的

Matroid flat counts are not unimodal

该研究针对罗塔1970年关于拟阵平坦计数序列是单峰的猜想给出反例，受拉尔森反驳梅森猜想启发，解释了一种能将平坦对数凹性失效转化为单峰性失效的机制。

秩二拟阵 Ehrhart $h^*$-多项式的实根性与超对数凹性

Real-rootedness and ultra log-concavity of rank-two matroid Ehrhart $h^*$-polynomials

本文证明秩二拟阵在特定条件下Ehrhart $h^*$-多项式的实根性，给出反例否证猜想，并证明超对数凹性及单峰性。

Hermite-Jensen极限与$q$-多项式$ d $ log-concavity

Hermite-Jensen limits and $d$ log-concavity of $q$-multinomials

研究$q$-多项式$ d $ log-concavity的性质，通过Hermite-Jensen极限和归一化Jensen多项式渐进行为，证明在特定条件下，$ q $-多项式满足更强的log-concavity不等式。

An overpartition analogue of $q$-binomial coefficients, II: combinatorial proofs and $(q,t)$-log concavity

In a previous paper, we studied an overpartition analogue of Gaussian polynomials as the generating function for overpartitions fitting inside an $m \times n$ rectangle. Here, we add one more parameter counting the number of overlined parts, obtaining a two-parameter generalization $\overline{m+n \brack n}_{q,t}$ of Gaussian polynomials, which is also a $(q,t)$-analogue of Delannoy numbers. First we obtain finite versions of classical $q$-series identities such as the $q$-binomial theorem and the Lebesgue identity, as well as two-variable generalizations of classical identities involving Gauss…

一种基于AI增强的树单峰猜想研究方法

An AI enhanced approach to the tree unimodality conjecture

本文利用AI架构PatternBoost训练机器寻找树独立集序列对数凹猜想的反例，成功找到大量顶点数27至101的新反例，并讨论了方法的成功与失败。

On some log-concavity properties of the Alexander-Conway and Links-Gould invariants

The Links--Gould invariant $\mathrm{LG}(L ; t_0, t_1)$ of a link $L$ is a two-variable quantum generalization of the Alexander--Conway polynomial $Δ_L(t)$ and has been shown to share some of its most geometric features in several recent works. Here we suggest that $\mathrm{LG}$ likely shares another of the Alexander polynomial's most distinctive - and mysterious - properties: for alternating links, the coefficients of the Links-Gould polynomial alternate and appear to form a log-concave two-indexed sequence with no internal zeros. The former was observed by Ishii for knots with up to 10 crossi…

关于交换置换的无限元组关于轨道数的对数凹性

Log-concavity with respect to the number of orbits for infinite tuples of commuting permutations

本文提出了关于交换置换组轨道数 $A(p,n,k)$ 关于 $k$ 的对数凹性猜想，并基于广义 Turán 数关于 $k$ 的对数凹性，证明了该猜想 $p=\infty$ 的情形。

满足某种超递推关系的三角阵列行的对数凹性

Log-concavity of rows of triangular arrays satisfying a certain super-recurrence

本文研究满足超递推关系的三角阵列，给出其行具有对数凹性的充分条件，统一证明多类经典组合序列的对数凹性并证实广义 Lah 数猜想，还引入双参数 Eulerian 数推广并讨论其性质。