Abstract
It is shown that the classical decomposition of permutations into disjoint cycles can be extended to more general mappings by means of path-cycles, and an algorithm is given to obtain the decomposition. The device is used to obtain information about generating sets for the semigroup of all singular selfmaps of $X_{n} = \{1, 2, \dots, n\}$. Let $T_{n,r} = S_{n}\cup K_{n,r}$, where $S_{n}$ is the symmetric group and $K_{n,r}$ is the set of maps $\alpha\,:\, X_{n} \to X_{n}$ such that $|im(\alpha)| \le r$. The smallest number of elements of $K_{n,r}$ which, together with $S_{n}$, generate $T_{n,r}$ is $p_{r}(n)$, the number of partitions of $n$ with $r$ terms.
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Ayik, G., Ayik, H. & Howie, J. On Factorisations and Generators in Transformation Semigroups. Semigroup Forum 70, 225–237 (2005). https://doi.org/10.1007/s00233-004-0145-x
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DOI: https://doi.org/10.1007/s00233-004-0145-x