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\centerline{\bf Math 365 -- Monday 3/25/19}


Recall:
$$
(1+x)^n = \sum_{k=0}^n \binom{n}{k} x^k, \qquad 
\frac{1 - x^{n}}{1-x} = \sum_{k=0}^{n-1} x^{k},$$
$$
e^x = \sum_{k=0}^\infty x^k/k!, \qquad \text{ and } \qquad \frac{1}{1-x} = \sum_{k=0}^\infty x^k$$
Also, for series 
$$f(x) = \sum_{k=0}^\infty a_k x^k \qquad \text{ and } \qquad  g(x) = \sum_{k=0}^\infty b_k x^k,$$ 
we have 
\begin{equation}\label{eq:series-ops}f(x) + g(x) = \sum_{k=0}^\infty (a_k + b_k) x^k
	\quad \text{ and } \quad 
	f(x) g(x) = \sum_{k=0}^\infty \left(\sum_{i = 0}^k a_i b_{k-i} \right) x^k.
\tag{$*$}
	\end{equation}

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\hrule

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\begin{try}Recall that the generating function for a sequence $\{a_0, a_1, a_2, \dots, \}$ is 
$$\qquad\qquad\qquad\sum_{k=0}^\infty a_k x^k. \qquad\qquad {{\text{(Considered ``formally'', i.e.\ without}\atop \text{consideration of convergence.)}}}$$
Give the generating functions, in terms of their series and in their simplified (closed) form, for each of the following sequences. In many cases, you'll use your answers from Exercise 36 (from last time). 
\begin{enumerate}[(a)]
\item $a_n = n+1$ for $n=0,1,2,\dots$. 

\medskip

\item $a_n = 3^n$, for $n=0, 1, \dots$.

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\item $a_n = \begin{cases} 1 & n = 3k \text{ for some $k\in\ZZ$},\\ 0, & n \neq 3k  \text{ for all $k\in\ZZ$},\end{cases}$  for $n=0, 1, \dots$.\\
(i.e.\ $a_n$ is 1 if $n$ is a multiple of $3$, and is 0 otherwise). 

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\item $a_n = 1/n$ for $n=1, 2,  \dots$.
\end{enumerate}
\end{try}


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\begin{try}
For each of the following recursively defined sequences, 
\begin{enumerate}[(i)]
\item solve using generating functions;
\item solve using the methods of Section 8.2 and compare to part (i);
\item check your answers by comparing the values you get by using your formula and by computing recursively for the first three terms of the sequence.
\end{enumerate}

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\begin{enumerate}[(a)]
\item $a_n = 2 a_{n-1} + 3$, $a_0  = 1$.

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\item $a_n = 3 a_{n-1} + 4^{n-1}$. 
\end{enumerate}
\end{try}

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\begin{try}~
\begin{enumerate}[(a)]
\item Consider the integer partitions of $4$.
\begin{enumerate}[(i)]
\item Write a corresponding generating function, wherein the coefficient of $x^4$ is the number of integer partitions of $4$.

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\item Make explicit the correspondence between the monomials in the expansion of your generating function and the integer partitions of $4$ (like we did for the partitions of 5 in class). 
\end{enumerate}

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\item Describe a combinatorial problem that is solved by calculating the coefficient of $x^6$ for the following generating functions.
\begin{enumerate}[(i)]
\item $(1 + x + x^2 + x^3)^4$

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\item $(1 + x + x^2 + \cdots + ) (1 + x^2 + x^4 + x^6) (1 + x^3 + x^6)$

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\item $(x + x^2 + x^3)^5$
\end{enumerate}

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\item For each of the following questions, give the corresponding generating function and determine $k$ such that the coefficient of $x^k$ is the answer the question.
\begin{enumerate}[(i)] 
\item How many different ways can 12 identical action figures can be given to five children so that each child receives at most three action figures?

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\item How many different ways 10 identical balloons can be given to four children if each child receives at least two balloons?

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\item How many different ways are there to choose a dozen bagels from three varieties--plain, onion, and raisin--if at least two bagels of each kind but no more than three plain bagels are chosen?

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\item How many ways can you make change for $\$100$ using $\$1$ bills, $\$5$ bills, $\$10$ bills, and $\$20$ bills?

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\end{enumerate}
\end{enumerate}
\end{try}

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\begin{try}(Generating functions for partitions)
\begin{enumerate}[(a)]
\item Write the generating function for partitions with even-sized parts, i.e.
$$\PART{4,4,2} \quad\text{ or } \quad\PART{8,6,4,2} \qquad \text{ but not } \qquad \PART{3,1} \quad\text{ or }\quad \PART{5,3,1}.$$
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\item Write the generating function for partitions with no more than two parts of each size, i.e.
$$\PART{4,4,2,2,1}\quad \text{ or }\quad \PART{5,3,1} \qquad \quad \text{ but not } \quad\qquad\PART{3,3,3,1} \text{ or } \PART{5,1,1,1}.$$
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\item Write the generating function for partitions parts all of prime size, i.e. 
$$\PART{3,3,2} \quad\text{ or } \quad\PART{5,3,2} \qquad \text{ but not } \qquad\PART{3,1} \quad\text{ or }\quad \PART{4,3,2}.$$
(The prime numbers are the integers $p$ greater than 1 that are divisible by 1 and $p$ but no other positive integers.)

\end{enumerate}
\end{try}


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