Tanky Woo >> 母函数(Generating function)详解 ― TankyWoo


Tanky Woo » 母函数(Generating function)详解 — TankyWoo
在数学中,某个序列的母函数(Generating function,又称生成函数)是一种形式幂级数,其每一项的系数可以提供关于这个序列的信息。使用母函数解决问题的方法称为母函数方法
母函数可分为很多种,包括普通母函数指数母函数L级数贝尔级数狄利克雷级数。对每个序列都可以写出以上每个类型的一个母函数。构造母函数的目的一般是为了解决某个特定的问题,因此选用何种母函数视乎序列本身的特性和问题的类型。
我们首先来看下这个多项式乘法:
母函数图(1)
由此可以看出:
1.x的系数是a1,a2,…an 的单个组合的全体。
2. x^2的系数是a1,a2,…a2的两个组合的全体。
………
n. x^n的系数是a1,a2,….an的n个组合的全体(只有1个)。

进一步得到:
母函数图(2)

母函数的定义
对于序列a0,a1,a2,…构造一函数:
母函数图(3)
称函数G(x)是序列a0,a1,a2,…的母函数。

这里先给出2个例子,等会再结合题目分析:

第一种:
有1克、2克、3克、4克的砝码各一枚,能称出哪几种重量?每种重量各有几种可能方案?
考虑用母函数来解决这个问题:
我们假设x表示砝码,x的指数表示砝码的重量,这样:
1个1克的砝码可以用函数1+1*x^1表示,
1个2克的砝码可以用函数1+1*x^2表示,
1个3克的砝码可以用函数1+1*x^3表示,
1个4克的砝码可以用函数1+1*x^4表示,
上面这四个式子懂吗?
我们拿1+x^2来说,前面已经说过,x表示砝码,x的指数表示砝码的重量!初始状态时,这里就是一个质量为2的砝码。
那么前面的1表示什么?按照上面的理解,1其实应该写为:1*x^0,即1代表重量为2的砝码数量为0个。
所以这里1+1*x^2 = 1*x^0 + 1*x^2,即表示2克的砝码有两种状态,不取或取,不取则为1*x^0,取则为1*x^2

不知道大家理解没,我们这里结合前面那句话:
把组合问题的加法法则和幂级数的乘幂对应起来

接着讨论上面的1+x^2,这里x前面的系数有什么意义?
这里的系数表示状态数(方案数)
1+x^2,也就是1*x^0 + 1*x^2,也就是上面说的不取2克砝码,此时有1种状态;或者取2克砝码,此时也有1种状态。(分析!)

所以,前面说的那句话的意义大家可以理解了吧?
几种砝码的组合可以称重的情况,可以用以上几个函数的乘积表示:
(1+x)(1+x^2)(1+x^3)(1+x^4)
=(1+x+x^2+x^4)(1+x^3+^4+x^7)
=1 + x + x^2 + 2*x^3 + 2*x^4 + 2*x^5 + 2*x^6 + 2*x^7 + x^8 + x^9 + x^10
从上面的函数知道:可称出从1克到10克,系数便是方案数。(!!!经典!!!)
例如右端有2^x^5 项,即称出5克的方案有2种:5=3+2=4+1;同样,6=1+2+3=4+2;10=1+2+3+4。
故称出6克的方案数有2种,称出10克的方案数有1种 。

接着上面,接下来是第二种情况:

第二种:
求用1分、2分、3分的邮票贴出不同数值的方案数:
大家把这种情况和第一种比较有何区别?第一种每种是一个,而这里每种是无限的。
母函数图(4)

以展开后的x^4为例,其系数为4,即4拆分成1、2、3之和的拆分方案数为4;
即 :4=1+1+1+1=1+1+2=1+3=2+2

这里再引出两个概念"整数拆分"和"拆分数":
所谓整数拆分即把整数分解成若干整数的和(相当于把n个无区别的球放到n个无标志的盒子,盒子允许空,也允许放多于一个球)。
整数拆分成若干整数的和,办法不一,不同拆分法的总数叫做拆分数
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