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, …, n}. Let T_(n,r) = S_n ∪ K_(n,r), where S_n is the symmetric group and K_(n,r) is the set of maps α: X_n → X_n such that |im(α)| ≤ 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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