How many compositions of $n \in N$ are there where each part is greater than $1$?

Question

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Let $n \in N$. How many compositions of $n$ are there where each part is greater than $1$? (Number of parts are not restricted)

Answer 1

We know that number of compositions of $n$ in $k$ positive parts is $$\binom{k-1}{n-1}$$ and to each composition of $n$ in $k$ parts correspondes a composition of $n+k$ into $k$ parts greater than $1$ if each part we increase by 1. Conversely: to each composition of n into k parts greater than two each part we decrease by 1 we get a composition of $n-k$ into k parts greater than 1 so number of compositions of $n$ into $k$ parts greater than $1$ is $$\binom{k-1}{n-k-1}$$ and the total number is $$\sum_{k\geq1}\binom{k-1}{n-k-1}=\sum_{k\geq1}\binom{k-1}{n-2-(k-1)}=F_{n-2}$$

$F_n$ is Fibonacci number