Combinatorial Enumeration and Random Generation

组合对象间双射的研究一直是数理组合学的研究热点之一。尽管它们具有广泛的适用性，但它们仍然完全缺乏形式上的理解，即哪些双射是等价的，哪些是“好的”，因为它们保持了组合对象的自然结构。我们启动了对分拆双射的渐近性质的研究，并声称一大类著名的双射实际上是“渐近稳定的”。利用CS式的归约思想，我们提出了第一种形式的方法来证明所有经典的Young表双射实际上是线性系等价的。还讨论了其他的组合对象和几个新的方向。从数学的早期开始，组合对象就扮演着重要的角色。这些对象包括各种图形、树、元素的划分、具有较小形状的区域的平铺等。寻找或估计这些对象的数量是一个基本问题，该问题在某些情况下已得到解决，在许多其他情况下仍未解决。有时，人们能够通过直接组合论元(双射)将这类对象的数量与其他对象的数量联系起来。我们建议深入研究这些双射的性质，是否(以及如何)它们不仅揭示了这些组合对象的数量，而且揭示了这些组合对象的结构结果。

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.