Automating Combinatorics

Doron Zeilberger建议继续开发利用计算机代数的巨大潜力进行组合学及相关领域研究的方法，并设计实验，（严格的）计算机辅助和计算机生成的研究。特别是他希望开发一种新的算法理论被命名为“符号矩演算”，这将产生计算算法，符号化和自动化，有趣的组合量的（统计）时刻。他还计划开发一个递归方程的一般理论，该理论将包括计算生物学中普遍存在的最大值的“动态规划”递归，以及mex运算，这发生在组合博弈理论中。他还建议继续他在“人工组合学”中的研究，并开发算法来发现和严格证明用于计数置换类的枚举方案，以及自动推导生成函数。另一个研究方向是自动行列式求值，它在应用数学和科学中有潜在的应用。这项研究应该是共生的，因为它的具体结果和基本方法都有望帮助计算机代数开发者改进和增强他们的系统。也希望这项研究将有助于实验数学的萌芽领域，因为它将有助于开发一种研究方法，用于进行计算机实验，输出严格（和有趣！）数学定理（和证明），而不仅仅是验证和阐述猜想。这项研究属于组合数学领域，其对科学和技术的有用性是众所周知的。特别是，计算机科学在很大程度上是基于组合学，电子通信和万维网也是如此。

Doron Zeilberger proposes to continue to developmethodologies for harnessing the great potential ofComputer Algebra to do research in Combinatorics andrelated areas, and design experiments for (rigorous)computer-assisted and computer-generated research.In particular he hopes to develop a new algorithmic theoryto be named `Symbolic Moment Calculus', that would producealgorithms for computing, symbolically and automatically, (statistical) moments of interesting combinatorial quantities.He also plans to develop a general theory of recurrence equationsthat would include both the `Dynamic Programming' recurrencesfeaturing the maximum, ubiquitous in Computational Biology,and the mex operation, that occurs in the theory ofCombinatorial Games.He also proposes to continue hisefforts in `Artificial Combinatorics', and develop algorithmsfor the discovery and rigorous proof of enumeration schemes for counting permutation classes, and for automatically deducing generating functions. Anotherline of research concerns automatic determinant evaluationsthat has potential applications in applied mathematics andscience.This research should be symbiotic, as it isexpected that both the concrete results and the underlying methodologies, would help computer algebra developersto improve and enhance their systems. It is also hoped thatthis research will contribute to the budding field of ExperimentalMathematics, in that it will help develop a research methodology forconducting computer experiments that output rigorous (and interesting!) mathematical theorems(and proofs), rather than just verifying and formulating conjectures.This research is in the field of Combinatorics, whose usefulnessto science and technology is well-known. In particular, computerscience is largely based on combinatorics, as is electroniccommunication and the World Wide Web.