Suppose you have n pairs of parentheses and you would like to form valid groupings of them, where “valid” means that each open parenthesis has a matching closed parenthesis. For example, “(()())” is valid, but “())()(” is not. How many groupings are there for each value of n? Perhaps a more precise definition of the problem would be this: A string of parentheses is valid if there are an equal number of open and closed parentheses and if you begin at the left as you move to the right, add 1 each time you pass an open and subtract 1 each time you pass a closed parenthesis, then the sum is always non-negative. Table 1 shows the possible groupings for 0 ≤ n ≤ 5.
假设有 n 对括号，您希望将它们组成有效的分组，其中 "有效 "的意思是每个开放括号都有一个匹配的封闭括号。例如，"（（）（）（））"是有效的，但"（）（）（）（）"不是。每个 n 值有多少个分组？也许对这个问题更精确的定义是这样的：如果一个括号字符串的开括号和闭括号的数量相等，并且从左边开始向右边移动，每经过一个开括号加 1，每经过一个闭括号减 1，那么总和总是非负的，那么这个括号字符串就是有效的。表 1 显示了 0 ≤ n ≤ 5 时可能的分组情况。
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{"title":"Stirling Numbers","authors":"E. Deza","doi":"10.1142/13463","DOIUrl":"https://doi.org/10.1142/13463","url":null,"abstract":"","PeriodicalId":260761,"journal":{"name":"Selected Chapters of Number Theory: Special Numbers","volume":"2 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-04-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121989486","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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