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数学代写|组合数学作业代写Combinatorial Mathematics代考|Generating Functions

如果你也在 怎样代写组合数学Combinatorial Mathematics这个学科遇到相关的难题,请随时右上角联系我们的24/7代写客服。组合数学Combinatorial Mathematics是数学的一个领域,主要涉及计数,作为获得结果的手段和目的,以及有限结构的某些属性。它与数学的许多其他领域密切相关,有许多应用,从逻辑学到统计物理学,从进化生物学到计算机科学。

组合数学Combinatorial Mathematics因其解决的问题的广泛性而闻名。组合问题出现在纯数学的许多领域,特别是在代数、概率论、拓扑学和几何学中,]以及在其许多应用领域。许多组合问题在历史上都是孤立考虑的,对某个数学背景下出现的问题给出一个临时的解决方案。然而,在二十世纪后期,强大而普遍的理论方法被开发出来,使组合学本身成为一个独立的数学分支。组合学最古老和最容易理解的部分之一是图论,它本身与其他领域有许多自然联系。在计算机科学中,组合学经常被用来获得算法分析中的公式和估计。

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数学代写|组合数学作业代写Combinatorial Mathematics代考|Generating Functions

数学代写|组合数学作业代写Combinatorial Mathematics代考|Ordinary Generating Functions

Let $a_{n}$ be the solution to a counting problem with $n$ as a parameter. A problem may have many parameters; at present we treat all but one as fixed and use the varying parameter as the index of a sequence. We associate a generating function in one variable with this counting sequence.
3.1.1. Definition. A formal power series is an expression of the form $\sum_{n=0}^{\infty} a_{n} x^{n}$. The (ordinary) generating function (OGF) for a sequence $\langle a\rangle$ of complex numbers is the formal power series $\sum_{n=0}^{\infty} a_{n} x^{n}$ or any expression having this formal power series expansion (also called generating series in the literature on formal power series).

When objects in a universe $U$ are assigned nonnegative integer values by a weight function $w$, with $a_{n}=|{u \in U: w(u)=n}|$, we say that the generating function for $\langle a\rangle$ is the enumerator by $w$ for $U$ or is indexed by $w$.
3.1.2. Example. Let $a_{n}$ be the number of binary lists of length $n$. The generating function for $\langle a\rangle$ is $\sum_{n=0}^{\infty} 2^{n} x^{n}$; it is the enumerator of binary lists, indexed by length. We say “indexed by length” because the values of the weight function index the terms in the sequence $\langle a\rangle$.

Next consider $k$-subsets of an $n$-set. We fix one parameter to form a generating function, writing $a_{k}=\left(\begin{array}{l}n \ k\end{array}\right)$. In $A(x)=\sum_{k=0}^{n}\left(\begin{array}{c}n \ k\end{array}\right) x^{k}$, the coefficient of $x^{k}$ is the number of $k$-subsets of $[n]$. Thus $A(x)$, or more properly $A_{n}(x)$, is the enumerator of subsets of $[n]$, indexed by size. Since the coefficient of $x^{k}$ in the expansion of $(1+x)^{n}$ is $\left(\begin{array}{l}n \ k\end{array}\right)$, we also say that $(1+x)^{n}$ is the generating function for $\langle a\rangle$.

In a problem with several parameters, it may not be obvious which is most useful as the index of summation in a generating function. We can also form a generating function in two variables: $\sum_{n=0}^{\infty} \sum_{k=0}^{\infty} a_{n, k} x^{n} y^{k}$.

数学代写|组合数学作业代写Combinatorial Mathematics代考|Coefficients and Applications

Formal power series expansions of generating functions can be manipulated much like ordinary power series in calculus, with the additional freedom of ignoring issues of convergence. (Niven [1969] presents a readable introduction to the theory.) When a function has a power series expansion at 0 , that is also its formal power series expansion. In Lemma 2.2.17 we expanded $(1-x)^{-n}$ using Taylor’s Theorem. In Theorem 3.1.10, the same expression emerged as the formal power series expansion of $(1-x)^{-n}$ using the combinatorial interpretation of formal power series products. This is no coincidence.

Equality holds because operations on formal power series agree with corresponding operations on power series that converge at numerical values. Since this holds for sum and product, statements like $A(x)^{m} A(x)^{n}=A(x)^{m+n}$ have the same proofs for formal power series as for power series.

To extend the correspondence, we define differentiation and “evaluation” for formal power series to agree with these operations on power series. In calculus, Taylor series are termwise differentiable, using uniform convergence of series of functions. For formal power series, this is the definition of differentiation. We will skip formal verifications, focusing instead on the use of generating functions.
3.2.1. Definition. When $A(x)$ is the generating function with expansion $A(x)=$ $\sum_{n=0}^{\infty} a_{n} x^{n}$, evaluation at 0 is defined by setting $A(0)=a_{0}$. The derivative of a formal power series $\sum_{n=0}^{\infty} a_{n} x^{n}$ is the formal power series $\sum_{n=1}^{\infty} n a_{n} x^{n-1}$; we write $A^{\prime}(x)$ for the derivative of a formal power series $A(x)$ in $x$.

Since the termwise behavior holds by definition, the sum and product rules for differentiation of formal power series are easy to verify (Exercise 30). Generally we have the following statement, which motivates writing $A^{\prime}(x)$ for the derivative of a formal power series $A(x)$ :
The derivative of the formal power series expansion of $A(x)$ is the formal power series expansion of the derivative of $A(x)$.
The occurrence of power series as formal power series is another excuse for our (ab)use of the word “function” in the term generating function. Some formal power series converge in a neighborhood of the origin when viewed as power series, but some do not. Evaluation of $A(x)$ at numerical values for $x$ (such as $x=1$ ) is allowed when the sum converges (such as when $A(x)$ is a polynomial in $x$ ).

数学代写|组合数学作业代写COMBINATORIAL MATHEMATICS代考|Exponential Generating Functions

We used OGFs in studying subset problems because products of formal power series have the proper effect on coefficients to model compound selection problems. For counting problems involving “labeled” objects, a different type of generating function plays the appropriate role.
3.3.1. Definition. The exponential generating function (EGF) for a sequence $\langle a\rangle$ is $\sum a_{n} x^{n} / n !$. An EGF is also called an exponential enumerator; an OGF is an ordinary enumerator.
3.3.2. Example. $k$-ary words, indexed by length. There are $k^{n}$ words of length $n$ consisting of letters from a set of size $k$. The EGF for $k$-ary words, enumerated by length, is $\sum_{n=0}^{\infty} k^{n} x^{n} / n !$. When $x$ is a number, this is the power series expression for $\mathrm{e}^{k x}$, where again $\mathrm{e}$ denotes the base of the natural logarithm. As discussed earlier for OGFs, the formal power series has the same behavior under addition and multiplication as the exponential function given by $\mathrm{e}^{k x}$. Hence we say that $\mathrm{e}^{k x} i$ is the exponential enumerator by length for $n$-ary words.

Words are the natural ordered analogue of multisets. An $n$-word uses a multiset of size $n$ from $[k]$, but the elements are chosen in order.

When $k=1$, we obtain $\mathrm{e}^{x}$, so the EGF for $k$-ary words is the product of $k$ copies of the EGF for 1-ary words; that is, $\mathrm{e}^{k x}=\left(\mathrm{e}^{x}\right)^{k}$. We need to understand the combinatorial meaning of the product to explain why the $k$-fold product enumerates the words from an alphabet of size $k$.

数学代写|组合数学作业代写Combinatorial Mathematics代考|Generating Functions

组合数学代写

数学代写|组合数学作业代写COMBINATORIAL MATHEMATICS代考|ORDINARY GENERATING FUNCTIONS

让一种n是一个计数问题的解决方案n作为参数。一个问题可能有很多参数;目前,我们将除一个以外的所有参数都视为固定参数,并使用可变参数作为序列的索引。我们将一个变量中的生成函数与这个计数序列相关联。
3.1.1。定义。形式幂级数是形式的表达∑n=0∞一种nXn. 这这rd一世n一种r是生成函数这GF对于一个序列⟨一种⟩复数的形式是幂级数∑n=0∞一种nXn或任何具有这种形式幂级数展开的表达式一种ls这C一种ll和dG和n和r一种吨一世nGs和r一世和s一世n吨H和l一世吨和r一种吨在r和这nF这r米一种lp这在和rs和r一世和s.

当宇宙中的物体在由权重函数分配非负整数值在, 和一种n=|在∈在:在(在)=n|,我们说生成函数为⟨一种⟩是枚举数在为了在或被索引在.
3.1.2。例子。让一种n是长度的二进制列表的数量n. 生成函数为⟨一种⟩是∑n=0∞2nXn; 它是二进制列表的枚举器,按长度索引。我们说“按长度索引”是因为权重函数的值对序列中的项进行索引⟨一种⟩.

接下来考虑ķ- 的子集n-放。我们固定一个参数来形成一个生成函数,写成一种ķ=(n ķ). 在一种(X)=∑ķ=0n(n ķ)Xķ, 的系数Xķ是数量ķ- 的子集[n]. 因此一种(X),或者更恰当地说一种n(X), 是子集的枚举数[n],按大小索引。由于系数Xķ在扩大(1+X)n是(n ķ),我们也说(1+X)n是生成函数⟨一种⟩.

在具有多个参数的问题中,可能不清楚哪个作为生成函数中的求和索引最有用。我们还可以在两个变量中形成一个生成函数:∑n=0∞∑ķ=0∞一种n,ķXn是ķ.

数学代写|组合数学作业代写COMBINATORIAL MATHEMATICS代考|COEFFICIENTS AND APPLICATIONS

生成函数的形式幂级数展开可以像微积分中的普通幂级数一样进行操作,并具有忽略收敛问题的额外自由。ñ一世在和n[1969]pr和s和n吨s一种r和一种d一种bl和一世n吨r这d在C吨一世这n吨这吨H和吨H和这r是.当一个函数在 0 处有幂级数展开时,这也是它的正式幂级数展开。在引理 2.2.17 中,我们扩展了(1−X)−n使用泰勒定理。在定理 3.1.10 中,同样的表达式出现为形式幂级数展开(1−X)−n使用正式电源系列产品的组合解释。这不是巧合。

等式成立,因为对形式幂级数的运算与对收敛于数值的幂级数的相应运算一致。由于这适用于 sum 和 product,因此语句如一种(X)米一种(X)n=一种(X)米+n形式幂级数的证明与幂级数的证明相同。

为了扩展对应关系,我们定义了形式幂级数的微分和“评估”,以与幂级数上的这些操作一致。在微积分中,泰勒级数是逐项可微的,使用函数级数的一致收敛。对于形式幂级数,这是微分的定义。我们将跳过形式验证,而是关注生成函数的使用。
3.2.1。定义。什么时候一种(X)是扩展的生成函数一种(X)= ∑n=0∞一种nXn, 0 处的评估由设置定义一种(0)=一种0. 形式幂级数的导数∑n=0∞一种nXn是正式的幂级数∑n=1∞n一种nXn−1; 我们写一种′(X)对于形式幂级数的导数一种(X)在X.

由于术语行为在定义上成立,形式幂级数的求和和乘积规则很容易验证和X和rC一世s和30. 通常我们有以下语句,它激发了写作一种′(X)对于形式幂级数的导数一种(X):
形式幂级数展开的导数一种(X)是导数的形式幂级数展开一种(X).
幂级数作为正式幂级数的出现是我们的另一个借口一种b在术语生成函数中使用“函数”一词。当被视为幂级数时,一些正式的幂级数会在原点附近收敛,但有些则不会。评估一种(X)在数值为X s在CH一种s$X=1$当总和收敛时允许s在CH一种s在H和n$一种(X一世s一种p这l是n这米一世一种l一世nx$)。

数学代写|组合数学作业代写COMBINATORIAL MATHEMATICS代考|EXPONENTIAL GENERATING FUNCTIONS

我们在研究子集问题时使用了 OGF,因为形式幂级数的乘积对模拟复合选择问题的系数具有适当的影响。对于涉及“标记”对象的计数问题,不同类型的生成函数起着适当的作用。
3.3.1。定义。指数生成函数和GF对于一个序列⟨一种⟩是∑一种nXn/n!. EGF 也称为指数枚举器;OGF 是一个普通的枚举器。
3.3.2. 例子。ķ-ary 单词,按长度索引。有ķn长词n由一组大小的字母组成ķ. EGF 为ķ-ary 单词,按长度枚举,是∑n=0∞ķnXn/n!. 什么时候X是一个数字,这是幂级数表达式和ķX, 又在哪里和表示自然对数的底。正如前面对 OGF 所讨论的,形式幂级数在加法和乘法下具有与由下式给出的指数函数相同的行为和ķX. 因此我们说和ķX一世是按长度计算的指数枚举数n-ary 词。

词是多重集的自然有序类似物。一个n-word 使用多组大小n从[ķ],但元素是按顺序选择的。

什么时候ķ=1, 我们获得和X, 所以 EGF 为ķ-ary 词是ķ1 进制单词的 EGF 副本;那是,和ķX=(和X)ķ. 我们需要了解产品的组合含义来解释为什么ķ-fold 产品从大小的字母中枚举单词ķ.

数学代写|组合数学作业代写Combinatorial Mathematics代考

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