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排列

下载 笔记本

排列，也称为“安排数”或“次序”，是有序列表

的元素的重新排列，使其与

自身形成一一对应。在一个包含

个元素的集合上的排列数由

给出（

阶乘； 1937, p. 18）。例如，

的排列，即

，以及

的排列，即

在语言当中能够借助以下命令去找到列表所存在的排列情况, 排列情况为排列。能够运用以下命令于语言中对长度为进行测试, 测试目标为该长度 , 该长度为 , 长度为。

的列表是否是 1, ...,

的排列。

(1977) 总结了好多生成排列的算法, 并且指出Heap在1963年提出的最小更改排列算法一般是最快的, 关于这一点在1990年第10页有表述。Séroul在2000年第213到218页, 给出了另外一种枚举排列的办法, Heap提出算法的时间是1963年。

从一个包含

个元素的集合中获得

个元素的有序子集的方式的数量由下式给出

( 1937, p. 18)，其中

是阶乘。例如，

的 2-子集，即

。包含

个元素的无序子集被称为给定集合的 k-子集。

把排列弄成置换环的乘积, 如此这般这般如此是唯一的（依照循环的顺序来说）。对于循环分解，有这样一个例子。

的排列。这表示为

其对应着不相交的置换环, 分别是(2)以及(143)。在选择循环分解的表示形式之际具有很大的自由度, 这是由于(1)循环是不相交的, 所以能够依照任何顺序去指定, 而且(2)给定循环的任何旋转所指定的都是相同的循环（1990, p. 20）。故而, (431)(2)、(314)(2)、(143)(2)、(2)(431)、(2)(314)以及(2)(143)均描述了相同的排列。

另一种表示法明确地标识了应用在

个元素上的排列之前和之后元素所占据的位置，它使用一个

矩阵，其中第一行是

第二排是不一样的排列形式喽, 比如说, 把数值为 1 跟数值为 2 的元素进行交换, 并且让数值为 3 的元素处于原来位置不动的那种排列, 是能够表述成。

对乘积而言, 任何排列也都是对换的, 排列一般是以字典序或者以对换序来进行表示的, 排列跟一对杨氏之间是存在对应关系的, 这种对应关系被称为对应, 排列通常以字典序或对换序表示。

个对象的错误排列数为

，其中

是最近整数函数。在

在多个有序对象里, 不存在任何对象处于其自身自然位置的那种排列, 被称作错位排列, 或者有的时候也叫完全排列, 而且这类排列所具有的数量是由次阶乘来确定的。

给出。

使用

其中

给出

因此，一次选择 0、1、... 或

个元素的方式的数量是

可以使用以下递归过程获得集合 1, ...,

的所有排列

思索那种不存在连续元素对呈现（也就是, 不存在上升或者下降的相邻连续情况）的排列方式。针对。

包含0、1、2……个元素, 这样的排列的数量是1、0、0、2、14、90、646、5242、47622……其中OEIS是相关序列。

设 整数 集合 1, 2, ...,

被排列，并且得到的序列被分成递增的游程。当

接近 无穷大 时，将第

个游程的平均长度表示为

, 前几个值被总结于下头的表格当中, 这里面的e是自然对数的底数, （Le在1983年, 第41 - 42页；Knuth于1998年）。

OEIS

近似值

1....

1.9524...

1.9957...

另请参见, 交错排列, 取球, 二项式系数, 选择, 圆排列, 组合, 错位排列, 欧拉数, 偶排列, k -子集, 线性扩展, 夫妻就座问题, 多重选择, 多项式系数, 多重集, 奇排列, 排列上升, 置换环, 排列倒置, 排列矩阵, 排列模式, 排列游程, 排列符号, 随机排列, 字符串, 次阶乘, 对换, 在课堂中探索此主题, 使用探索。

被引用排列在 上的情况如下, 有在纽约的 中存在的 §10.1 , 其年份为1942 , 页码是239 - 240 ；还有列于 Le 著, 在巴黎出版且年份为1983的 中；还有 R.所著的内容, 其在 > 关于 的内容 中, 网址为~cos/inf/perm/.html ；还有被 的 中引用的 R。其出版年份为1977 , 页码范围是137 - 164 ；还有 Séroul 所著的 中关于 's' 的内容, 位于对于 的 中的 §8.15 , 其年出版于 , 页码是213 - 218 , 年份为2000；还有 S.所著的内容, 在关于 和带 的图的 中的 §1.1 , 位于马萨诸塞州的 出版的 中, 页码范围是3 - 16 , 年份为1990；还有 N. J. A. 所涉及的 /M1675 、/M2070 以及 在关于“ 的在线 ”中的情况；还有 H. F.所著的 “Perm ( 115)” 位于《计算机协会通讯》第5卷, 页码是434 - 435 , 年份为1962；还有 J. V. 被至 所引用, 书在纽约由 -Hill 出版, 页码为18 , 年份为1937。被引用排列请引用为。Csdn.tphep.cN/Article/details/70473.shtml
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