初探卡特兰数及有关问题(On Cartland number and related problems)

星期日,哥参加了上大学以来的第一次计算导论与程序设计的上机考试,可是最后一道题没AC。

这道题给了卡特兰数的一种通项公式,让你求卡特兰数的第n项。

从考场走出来之后,心里空落落的,不仅因为这道题没打出来直接影响了整个考试,还因为自己似乎从来没完全出于兴趣研究过某个数学问题……

这道题被我AC了后,我上网小查了一下卡特兰数的有关知识,发现卡特兰数还和几种类型的问题联系紧密。
所以,我觉得有必要在此小小的研究一下神奇的卡特兰数~

卡特兰数的前几项以及其递推公式,是大家对卡特兰数最熟悉的地方,我们不妨从这里开始对它的研究。

一、初识卡特兰数

首先,我们给出卡特兰数的前10项:

第0项:1
第1项:1
第2项:2
第3项:5
第4项:14
第5项:42
第6项:132
第7项:429
第8项:1430
第9项:4862

然后,我们给出卡特兰数的几种通项公式:

第一种:\({C_{n + 1}} = \frac{{2(2n + 1)}}{{n + 2}}{C_n}\)

第二种:\({C_n} = {C_0}{C_{n – 1}} + {C_1}{C_{n – 2}} + \cdots + {C_{n – 2}}{C_1} + {C_{n – 1}}{C_0}(n \ge 2)\)

第三种:\({F_n} = \frac{{C_{2n}^n}}{{n + 1}}\) (用\(F_n\)是为了和组合数公式\(C_{2n}^n\)区分)

第四种:\({C_n} = \prod\limits_{k = 2}^n {\frac{{n + k}}{k}}\)

二、卡特兰数的发现与定义

卡特兰数(英语:Catalan number),又称卡塔兰数、明安图数,是组合数学中一种常出现于各种计数问题中的数列。以比利时的数学家欧仁·查理·卡特兰的名字来命名。1730年左右被蒙古族数学家明安图使用于对三角函数幂级数的推导而首次发现,1774年被发表在《割圜密率捷法》。

————百度百科

卡特兰数的定义:

There are many, many combinatorial definitions of the Catalan numbers, but the most common might be that Cn counts the number of lattice paths from (0,0) to (n,n) that only take unit step right and up, and never cross the diagonal line y=x (but they are allowed to touch the diagonal line). There is not a unique definition of Catalan numbers because all of the various combinatorial definitions are equivalent to each other, so which one you take as your definition is a stylistic preference.
卡特兰数的组合定义很多,但是最常见的定义也许是从点(0,0)走到(n,n),只向右向上走且不越过对角线的路径的数目。卡特兰数没有唯一的定义,因为所有不同的组合定义都是相互等价的,所以哪个是你心目中的定义方式,就看你自己了。
*

————Math StackExchange论坛,源地址

————————

On Sunday, I took the first computer test of introduction to computing and programming since I went to college, but I didn’t have AC in the last question.

This problem gives a general term formula of Cartland number, which allows you to find the nth term of Cartland number.

After coming out of the examination room, I felt empty, not only because this question didn’t type out, which directly affected the whole exam, but also because I never seemed to have studied a mathematical problem completely out of interest

After this problem was solved by me, I checked the relevant knowledge of Cartland number on the Internet and found that Cartland number is also closely related to several types of problems.
Therefore, I think it is necessary to study the magical Cartland number here~

The first few terms of Cartland number and its recurrence formula are the most familiar places for us to study Cartland number. We might as well start to study it here.

1、 First knowledge of Cartland number

First, we give the first 10 terms of Cartland number:

Item 0: 1
Item 1: 1
Item 2: 2
Item 3: 5
Item 4: 14
Item 5: 42
Item 6: 132
Item 7: 429
Item 8: 1430
Item 9: 4862

Then, we give several general term formulas of Cartland number:

The first one: \ ({C {n + 1}} = \ frac {2 (2n + 1)} {{n + 2} {C} \)

The second type: \ ({c_n} = {c_0} {c_ {n – 1} + {c_1} {c_ {n – 2}} + \ cdots + {c_ {n – 2} {c_1} + {c_ {n – 1}} {c_0} (n \ Ge 2) \)

The third type: \ ({f _n} = \ frac {C _ {2n} ^ n} {n + 1} \) (use \ (f _n \) to distinguish it from the combinatorial number formula \ (C _ {2n} ^ n \)

The fourth type: \ ({C _n} = \ prod \ limits {k = 2} ^ n {\ frac {{n + K} {K} \)

2、 Discovery and definition of Cartland number

< strong > Catalan number (English: Catalan number), also known as Catalan number and mingantu number, is a number sequence that often appears in various counting problems in combinatorics. Named after the Belgian mathematician Oren Charlie Cartland. It was first discovered by Mongolian mathematician Ming Antu in the derivation of trigonometric function power series around 1730, and was published in the shortcut method of cutting circle density in 1774

————Baidu Encyclopedia

Definition of Cartland number:

There are many, many combinatorial definitions of the Catalan numbers, but the most common might be that Cn counts the number of lattice paths from (0,0) to (n,n) that only take unit step right and up, and never cross the diagonal line y=x (but they are allowed to touch the diagonal line). There is not a unique definition of Catalan numbers because all of the various combinatorial definitions are equivalent to each other, so which one you take as your definition is a stylistic preference.
卡特兰数的组合定义很多,但是最常见的定义也许是从点(0,0)走到(n,n),只向右向上走且不越过对角线的路径的数目。卡特兰数没有唯一的定义,因为所有不同的组合定义都是相互等价的,所以哪个是你心目中的定义方式,就看你自己了。
*

————Math stackexchange forum, source address