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\frametitle{Partitions of Integers}

Grimaldi 9.3

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In number theory the function $p(n)$ counts the number of ways to partition
a positive integer $n$ into positive summands.

\begin{tabular}{r|l}
$n$ & $p(n)$ \\ \hline 
1   &   1  \\
2 = 1+1 & 2 \\
3 = 2+1 = 1+1+1 & 3 \\
4 = 3+1 = 2+2 = 2+1+1 = 1+1+1+1 & 5 \\                
                &  \\
                &  \\
\end{tabular}

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So the generating function for $p(n)$ is

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$P(x) = 0 + 1 x + 2 x^2 + 3x^3 + 5x^4 +  $
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Example.  Find the GF for the number of ways an advertiser can buy
$n$ minutes of air time if commercials come in 15, 30, or 60 second blocks.

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Exercise.  Find the GF for $d(n)$ the number of partitions of $n$ into 
{\bf distinct summands}.  E.g. $d(3)=2$ as $3=1+2$.  What is $d(5)$?

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