Consider the set $1, 2, \dots, N$, where $N>1$ is a natural number. In general, there are $N!$ permutations of this list. Let $\sigma$ be one such permutation. We define the tuple $\varphi(\sigma) = \left (c_1,\dots,c_N \right)$, where $c_i$ is the number of cycles of length $i$ in $\sigma$. This...