The mysterious story of square ice, piles of cubes, and bijections
The mysterious story of square ice, piles of cubes, and bijections
复制标题
方冰、立方体堆和双射的神秘故事
DOI：
10.1073/pnas.2005525117
复制
发表时间：
2020
期刊：
影响因子：
--
通讯作者：
Matjaž Konvalinka
中科院分区：
文献类型：
--
作者：
Ilse Fischer;Matjaž Konvalinka
Significance In the early 1980s, it was discovered that alternating sign matrices (ASMs), which are also commonly encountered in statistical mechanics, are counted by the same numbers as two classes of plane partitions. Since then it has been a major open problem in this area to construct explicit bijections between the three classes of objects. In this paper we describe a bijective proof that relates ASMs to one of the two classes of plane partitions as well as a bijective proof of the fact that the numbers can be expressed by a simple product formula. The constructions are complicated, but they are completely explicit and also provided in the form of computer code. When combinatorialists discover two different types of objects that are counted by the same numbers, they usually want to prove this by constructing an explicit bijective correspondence. Such proofs frequently reveal many more details about the relation between the two types of objects than just equinumerosity. A famous set of problems that has resisted various attempts to find bijective proofs for almost 40 y is concerned with alternating sign matrices (which are equivalent to a well-known physics model for two-dimensional ice) and their relations to certain classes of plane partitions. In this paper we tell the story of how the bijections were found.
