Schur正性与Narayana多项式的$q$-对数凸性

通过分配格实现的对数凹性与对数凸性

Log-concavity and log-convexity via distributive lattices

本文提出序理想引理，通过分配格中的序理想统一证明多种组合序列的对数凹性与对数凸性，并给出猜想与未来方向。

Parabolic Catalan numbers count flagged Schur functions; Convexity of tableau sets for Demazure characters

Shuffles are n-multipermutations with suit multiplicities given by a subset R of {1,..,n-1}. Their inverses are ordered partitions of {1,..,n} whose block sizes derive from R. These "R-permutations" depict the min length coset reps for the quotient of S_n by the parabolic subgroup W_J, with J the complement of R. We refer to those that blockwise avoid the pattern 312 as "312-avoiding R-permutations" and define the "parabolic R-Catalan number" to be the number of them. Let lambda be a partition of N with at most n parts whose set of shape column lengths less than n is R. We show that the number…

Positivity of iterated sequences of polynomials

In this paper, we present some criteria for the $2$-$q$-log-convexity and $3$-$q$-log-convexity of combinatorial sequences, which can be regarded as the first column of certain infinite triangular array $[A_{n,k}(q)]_{n,k\geq0}$ of polynomials in $q$ with nonnegative coefficients satisfying the recurrence relation $$A_{n,k}(q)=A_{n-1,k-1}(q)+g_k(q)A_{n-1,k}(q)+h_{k+1}(q)A_{n-1,k+1}(q).$$ Those criterions can also be presented by continued fractions and generating functions. These allow a unified treatment of the $2$-$q$-log-convexity of alternating Eulerian polynomials, $2$-log-convexity of Eu…

对偶$k$-Schur多项式的牛顿多面体

Newton polytopes of dual $k$-Schur polynomials

本文通过递归生成半标准$k$-表的算法，证明以$k$-有界分拆为指标的对偶$k$-Schur多项式与同指标Schur多项式支撑集一致、共享饱和牛顿多面体，进而得到三类多项式的M-凸性。

强 $q$-对数凸性的 $d$-Hoggatt 多项式

Strong $q$-log-convexity of the $d$-Hoggatt polynomials

本文研究 $d$-Hoggatt 数的对数凸性，证明 $d$-Hoggatt 变换保持对数凸性，建立 $q$-模拟的强 $q$-对数凹性，并基于 Schur 函数理论证明 $d$-Hoggatt 多项式序列强 $q$-对数凸。

A型Demazure特征标（关键多项式）的表集合凸性，抛物Catalan数

Convexity of tableau sets for type A Demazure characters (key polynomials), parabolic Catalan numbers

本文证明A型Demazure特征标对应的半标准杨表集合的凸性等价于R-312-避免条件，并定义R-抛物Catalan数，为后续深化Reiner-Shimozono及Postnikov-Stanley结果奠定基础。

泊车函数、斯米尔诺夫字与非交叉Chow多项式

Real-rootedness and interlacing for parking functions and Chow polynomials

该研究通过转换泊车函数与斯米尔诺夫字的关系，结合递推关系证明了非交叉分拆格Chow多项式等的实根性，解决了相关猜想并得到多个多项式的实根性结论。

Linear transformations and strong $q$-log-concavity for certain combinatorial triangle

It is well-known that the binomial transformation preserves the log-concavity property and log-convexity property. Let $\binom{a+n}{b+k}$ be the binomial coefficients and $\binom{n,k}{j}$ be defined by $(b_0+b_1x+\cdots+b_kx^{k})^n:=\sum_{j=0}^{kn}\binom{n,k}{j}x^j,$ where the sequence $(b_i)_{0\leq i\leq k}$ is log-concave. In this paper, we prove that the linear transformation $$y_n(q)=\sum_{k=0}^n\binom{a+n}{b+k}x_k(q)$$ preserves the strong $q$-log-concavity property for any fixed nonnegative integers $a$ and $b$, which strengthens and gives a simple proof of results of Ehrenborg and Stein…

对数凸性导出的优超不等式

Majorization Inequalities from Logarithmic Convexity

本文利用对数凸性作为统一工具，证明Macdonald多项式、Jack多项式和Heckman-Opdam超几何函数的新优超不等式，统一现有结果并解决若干开放猜想。

加权键偏序集与一种新的色对称函数

Weighted bond posets and a new chromatic symmetric function

该研究利用加权键格引入图的新多项式与对称函数不变量，证明弦图下的多项式$γ$-正性与最高次齐次分量$e$-正性，并提出Schur-对数凹猜想。

基于洛伦兹多项式的相交理论不等式

Intersection theoretic inequalities via Lorentzian polynomials

该研究将洛伦兹多项式应用于多类几何领域，建立了rKT性质等相交理论不等式与凸性几何不等式，拓展了多拟阵结构适用范围并得到加性组合不等式的几何类似结果。

L-凹性与Lam、Postnikov和Pylyavskyy猜想的证明

L-log-concavity and a proof of the conjecture of Lam, Postnikov and Pylyavskyy

本文通过引入新的组合模型`skeps`和利用Murota的L-凸性理论，证明了Lam、Postnikov和Pylyavskyy猜想，展示了s_{λ'}s_{μ'} - s_λs_μ的Schur非负性。

Plethysm 系数的稳定性与修正多项式归纳

Stability of plethysm coefficients and modified polynomial induction

本文给出 plethysm 系数两个稳定性结果的简短证明，并建立稳定值与 Weyl 模重数的新联系，提出稳定 Foulkes 系数的向量划分函数公式。