\(\mathfrak{S}_n\) is amazing (part 1)

Let \(\mathfrak{S}_n\) denote the set of all permutations of the set \([n]=\{1,2,3 \cdots n\}\).

Any permutation \(w\in \mathfrak{S}_n\) can be written in what is called cycle notation, described here. This can be made a unique representation by requiring that the maximum element of each cycle goes first and that cycles are ordered in increasing order by their largest elements. This is known as the standard representation of a permutation. As an example, in \(\mathfrak{S}_4\) the permutation \(2314\) has a standard representation of \((312)(4)\).

The map \(\phi_n: \mathfrak{S}_n \to \mathfrak{S}_n\) takes a permutation \(w\) and maps it to its standard representation without the parentheses. For how many permutations \(w\in \mathfrak{S}_{10}\) does \(w= \phi_{10}(w)\)?

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