The monoid of all maps on $n$ letters is denoted $T_n$ and called the full transformation monoid. Your intuition is both right and wrong. The irreducible representations of $T_n$ are in bijection with irreducible representations of all symmetric groups of degree at most $n$. The character table is block upper triangular with diagonal blocks character tables of symmetric groups of degree at most $n$. Thus inverting the table, which is how you decompose characters into irreducibles, is a sort of generalized inclusion-exclusion. Putcha computed the character table, although Hewitt and Zuckermann stated a portion of the result without a complete proof in 1957. But this is only the semisimple part of the story.
The algebra of $T_n$ is not semisimple. For example, nobody knows what the projective indecomposables look like in general. The algebra is quasihereditary and so has finite global dimension. Putcha computed a lot of information about the quiver. Ponizovsky showed $T_1,T_2,T_3$ has finite representation type, Ringel showed $T_4$ has finite representation type and Putcha showed the representation type is infinite in all other cases. I think this part of the representation theory is not completely controlled by symmetric groups.
Update 1/15.
In case you are still interested I just put up a paper
http://arxiv.org/abs/1502.00959 computing the global dimension of this algebra. From the paper you can get a very precise idea of how the representations of the symmetric groups of different degrees interact to control the Ext spaces between simple modules in a non trivial way. Also the paper gives a more accurate survey of the representation theory of this algebra than my answer.
