Combinatorial Enumeration and Random Generation

组合枚举和随机生成

• 批准号: 0402028
• 负责人: Igor Pak
• 金额: --
• 依托单位: Massachusetts Institute of Technology
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2004
• 资助国家: 美国
• 起止时间: 2004-07-01 至 2008-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0402028&HistoricalAwards=false
• 关键词: Combinatorial; Enumeration; Random; Generation

The study of bijections between combinatorial objectshas always remained an attractive part of Enumerativeand Algebraic Combinatorics. Despite their wideapplicability, their remains a complete lack of formalunderstanding as to which bijections are equivalent,and which one are "good" in a sense that they preservethe natural structure of the combinatorial objects.We initiate the study of asymptotic properties ofpartition bijections and claim that a large class ofwell known bijections are in fact "asymptotically stable".Using CS-style reduction ideas we propose the firstformal way to formulate that all classical Young tableaubijections are in fact linear tie equivalent. Othercombinatorial objects and several new directionsare also discussed. Since the early days of mathematics, combinatorial objectshave played an important role. These objects, whichinclude various sets of graphs, trees, partitions ofintegers, tilings of regions with smaller shapes, etc.Finding or estimating the number of such objects is afundamental problem which has been resolved in someinstances and remains open in many other case. Sometimesone is able to relate the number of such objects to thenumber of other objects by means of a direct combinatorialargument (a bijection). We propose an in-depth study ofthe nature of these bijection, as to whether (and how)they reveal not just the number, but structural resultson these combinatorial objects.

对组合对象SHA之间的射击的研究始终是枚举代数组合学的有吸引力的一部分。 尽管它们具有宽大的性能，但它们仍然完全缺乏形式化的射击，而对于哪个意义上的，哪个是“好”的，因为它们保留了组合物体的自然结构。我们启动了分散生物的渐进性研究的研究，并声称一大型已知的已知的双子都在构想的是“毫无疑问”构想的始终稳定范围。表明所有经典的年轻tableaubijections实际上都是线性扎带的。 还讨论了其他combinatorial对象和几个新方向。自数学早期以来，组合对象旋转起着重要作用。 这些对象，包括各种图形，树，分区的分区，形状较小的区域的瓷砖等。发现或估计此类对象的数量是在某些情况下解决并在许多其他情况下仍在开放的问题。 有时，一个人能够通过直接组合物测量（两次射击）将此类对象的数量与其他对象相关联。 我们提出了一项对这些生物的性质的深入研究，即（以及如何）它们是否不仅揭示了数量，而是结构性结果，而是这些组合对象。