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{
\bf  Quiz \# 10 for Dr. Z.'s Number Theory
}
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{\bf NAME:} (print!) $\underline
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{\bf E-MAIL ADDRESS:} (print!) 
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{\bf 1.} An {\it extremely distinct} partition
of $n$ is a sequence of integers
$$
(\lambda_1, \lambda_2, \dots, \lambda_t) \quad ,
$$
such that
$$
\lambda_1 + \lambda_2 + \dots + \lambda_t = n \quad ,
$$
and
$$
\lambda_1 - \lambda_2 \geq 2
\quad
\lambda_2 - \lambda_3 \geq 2
\quad ,
\dots ,
\quad,
\lambda_{t-1} - \lambda_t \geq 2
$$
and
$$
\lambda_t>0 \quad,
$$
Let $q(n)$ be the number of 
partitions of $n$, and $q(n,k)$ 
be the number of exteremely distinct partitions of 
$n$ whose largest part is $k$.

(i) (5 points) Explain why
$$
q(n,k) = \sum_{r=1}^{k-2} q(n-k, r) \quad ,
$$
and, of course
$$
q(n,n)=1 \quad .
$$
(ii) Use the above recurrence, and 
$$
q(n)=\sum_{k=1}^{n} q(n,k)
$$ 
to 
compute $q(n)$ for $1 \leq n \leq 5$.





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