For a finite group $G,$ $\mathsf{D}(G)$ is defined as the least positive integer $k$ such that for every sequence $S=g_1\bdot g_2\bdot \dotsc \bdot g_k$ of length $k$ over $G$, there exist $1 \le i_1 < i_2 <\cdots < i_m \le k $ such that $g_{i_1}g_{i_2}\cdots g_{i_m}=1,$ where $1$ is the identity element of $G.$ The small Davenport constant $\mathsf{d}(G)$ is the maximal positive integer $k$ such that there is a sequence of length $k$ over $G$ which has no non-trivial product-one subsequence. In 2004, Dimitrov proved that $\mathsf{D}(G)\leq \mathsf{L}(G)$ for a finite $p$-group $G$, where $p$ is a prime and $\mathsf{L}(G)$ is the Loewy length of $\mathbb{F}_p[G].$ He conjectured that the equality holds for all finite $p$-groups. In this article, we compute $\mathsf{D}(G)$ for certain classes of finite non-abelian $p$-groups, including metacyclic groups, and show that the conjecture is true by determining the precise value of $\mathsf{L}(G)$. As a consequence, we refine an upper bound on $\mathsf{d}(G)$ recently given by Qu, Li and Teeuwsen, and prove that for specific classes of groups $\mathsf{D}(G)=\mathsf{d}(G)+1$. We also evaluate $\mathsf{D}(G)$ for finite dicyclic, semi-dihedral and other groups.
Combinatorial invariants for certain classes of non-abelian groups
Indian Institute of Science Education and Research Bhopal（印度科学教育研究学院博帕尔分校）；Technische Universität Braunschweig（布伦瑞克工业大学）；Ahmedabad University（艾哈迈达巴德大学）
Sharp Bounds on Davenport-Schinzel Sequences of Every Order
计算几何中一个长期存在的未解决问题是确定$n$单变量函数的下包络，对于某个固定的$s$，每对单变量函数最多交叉$s$次。已知这个问题等价于限定顺序$s$ Davenport-Schinzel序列的长度，即，在$n$字母表上的序列，其避免了形式$a \cdots b \cdots a \cdots b \cdots$与长度$s+2$的交替连续。这些序列是由达文波特和Schinzel在1965年引入的，用于模拟微分方程中的某个问题，并已被应用于限制几何算法、数据结构和几何排列的组合复杂性的运行时间。令$λ_s(n)$为顺序$s$ DS序列在$n$字母上的最大长度.什么是$λ_s$渐近？当$s$是偶数或$s\le 3$时，这个问题已经得到了满意的回答（Hart和Sharir，Agarwal，Sharir和Shor，Klazar和Nivasch）。然而，自从20世纪80年代中期阿加瓦尔、沙里尔和肖尔的工作以来，我们对奇数阶的理解一直存在着持续的差距。在这项工作中，我们有效地关闭的问题，建立尖锐的界限Davenport-Schinzel序列的每一个顺序$s$。我们的研究结果表明，与人们的直觉相反，当$s$是奇数时，$λ_s(n)$的行为基本上像$λ_{s-1}(n)$。这反驳了Alon等人（2008）和Nivasch（2010）的观点。
Three proofs of the Goulden-Litsyn-Shevelev conjecture on a sequence arising in algebraic geometry
I. P. Goulden，S. Litsyn和V. Shevelev [On a sequence arising in algebraic geometry，J. J. Sequences 8（2005），05.4.7]证明了与函数方程的解相关联的某些Laurent多项式仅具有奇数负幂。我们证明了他们的猜想并推广了它。
On a conjecture of Kolokolnikov on algebraic connectivity
School of Mathematical Sciences, Shanghai Jiao Tong University（上海交通大学数学科学学院）；College of Mathematics and Statistics, Hunan Normal University（湖南师范大学数学与统计学院）；School of Mathematical Sciences, East China Normal University（华东师范大学数学科学学院）
