第二类Stirling数的单峰型问题的研究

项目名称： 第二类Stirling数的单峰型问题的研究

项目编号： No.11201260

项目类型： 青年科学基金项目

立项/批准年度： 2013

项目学科： 数理科学和化学

项目作者： 刘丽

作者单位： 曲阜师范大学

项目金额： 22万元

中文摘要： 组合序列单峰型问题的研究是目前组合数学中的热点课题之一。第二类Stirling数S(n,k)是最基本的一类组合序列，研究成果极为丰富，然而仍存在许多公开问题。本项目拟结合组合、代数和分析方法研究第二类Stirling数的单峰型问题，主要研究内容包括： 1.Wegner猜想。熟知S(n,k)关于k是单峰的且至多有两个峰点，Wegner猜想当n＞2时峰点是唯一的。本项目将考虑n充分大时峰点的唯一性问题。 2.Engel猜想。S(n,k)计数了n元有限集有k个块的分拆数目，Engel猜想这个有限集上的所有分拆所含块数的平均值是凹的，Canfield证明了n充分大时猜想是对的。本项目将致力于改进目前的结果。 3.Wilf猜想。Wilf猜想当n＞2时，Bell多项式不以-1为零点。换言之，n元有限集的所有分拆中，含有偶数个块数的分拆数目不等于含有奇数个块数的分拆数目。本项目冀望完全证明这个猜想。

中文关键词： 零点相容性；连分式；强q-对数凸性；单峰型性质；第二类Stirling数

英文摘要： Unimodality problem of combinatorial sequences is one of hot topics in combinatorics. The Stirling number of the second kind is a primary combinatorial sequence. It has strong combinatorial background and related results are pretty rich. But now there are also many open problems. Using algebraic, combinatorial and analysis methods, we mainly focus on unimodality problems of Stirling numbers of the second kind in this research, including: 1.Wegner's conjecture. It is well known that Stirling numbers of the second kind are unimodal with at most two modes. When n＞2, Wegner conjectured that the mode of Stirling numbers of the second kind is unique. We will consider this conjecture for sufficiently large n. 2.Engel's conjecture. The Stirling number of the second kind is the number of partitions of an n-set having exactly k blocks. Engel conjectured that the average number of blocks in a partition of an n-set is concave. Canfield proved that it is true when n is sufficiently large. We try to improve existing results about this conjecture. 3.Wilf's conjecture. Wilf conjectured that -1 is not a zero of the Bell polynomials for n＞2. In other words, the number of partitions of an n-set with even-numbered blocks is not equal to that with odd-numbered blocks. We will try to prove this conjecture completely.

英文关键词： Compatible property；Continued fraction；Strong q-log-convexity；Unimodality property；The Stirling number of the second kind