形状为(n,n)标准杨表的一个双射  

A bijection on standard Young tableaux of shape(n,n)

作　　者：胡莹莹[1] 熊义财 孙毅[1] HU Yingying;XIONG Yicai;SUN Yi(College of Mathematics and System Science,Xinjiang University,Urumqi 830046,China)

机构地区：[1]新疆大学数学与系统科学学院,乌鲁木齐830046

出　　处：《黑龙江大学自然科学学报》2022年第1期24-29,共6页Journal of Natural Science of Heilongjiang University

基　　金：Supported by the National Natural Science Foundation of China(20190801245,11701491);the National University Student Innovation Program。

摘　　要：卡塔兰数在组合学和概率论中无处不在。首先利用组合学中标准杨表的钩长公式发现形状为(n,n)的标准杨表的个数恰好是卡塔兰数。借助RSK算法发现,至多只有两行的同形状的成对标准杨表的个数也是卡塔兰数。因此,这两个集合之间必定存在某种一一对应关系。本文将给出一个这样的一一映射。It is well known that Catalan numbers are ubiquitous in Combinatorics.It is found that all the standard Young tableaux of shape(n,n)are counted by Catalan numbers via the hook-length formula.Using the RSK algorithm,it can be seen that the set of all pairs of standard Young tableaux with at most two rows are also counted by Catalan numbers.This shows that there must be a bijection between the set of all the standard Young tableaux of shape(n,n)and the set of all pairs of standard Young tableaux with at most two rows.Such a bijection between these two sets will be provided.

关 键 词：一一映射 卡塔兰数 杨图 标准杨表 

分 类 号：O175[理学—数学]