Explicit description of all morphisms between symmetric groups. There is a well-known morphism $S_4\to S_3$, obtained by having $S_4$ act on the three partitions of $4$ objects into $2+2$. Similarly, given any $n$, one can devise a morphism $S_n\to S_k$ for some $k$ by having $S_n$ act on the partitions of $n$ objects into 
$n_1+n_2+\ldots+n_\ell$. One can further endow some or all of the element of the partition with an order, for example with $\ell=1$ one has $S_n$ acting on the set of ordered sequences of size $n$, and gets the left action of $S_n$ on itself, which is a morphism $S_n\to S_{n!}$ (the largest one that is irreducible).
Is that, are something close to that, the complete list of all morphisms $S_n\to S_k$ (up to conjugacy, of course)? I assume the answer is well-known.
Edit: the question was really naive, but I would like to know if some general information are nevertheless available on the Burnside ring of $S_n$ (which encodes the permutation representations of a finite group, but this is almost all I know).
 A: Bret Benesh and Ben Newton determined all pairs $(m,n)$ such that $S_m$ contains a maximal subgroup isomorphic to $S_n$. They are either $(n+1,n)$ with the obvious inclusion (or mapping $S_5$ into the image of a point stabilizer under the outer automorphism of $S_6$); $(\binom{n}{k},n)$, coming from the action of $S_n$ on the subsets of $k$ elements of $\{1,2,\ldots,n\}$; and $((kr)!/(r!)^k k!, kr)$ with $1\lt k,r$, with $S_{kr}$ acting on the the right cosets of a maximal subgroups of the wreath product $S_k\wr S_r$. This appears in A classification of certain maximal subgroups of symmetric groups, J. Algebra 304 (no. 2) pp. 1108-1113, MR2265507.
Bret later also determined all pairs $(m,n)$ such that $S_m$ has a maximal subgroup isomorphic to $A_n$; such that $A_m$ has a maximal subgroup isomorphic to $S_n$; and such that $A_m$ has a maximal subgroup isomorphic to $A_n$. This appears in the book Computational Group Theory and the Theory of Groups, Contemporary Mathematics 470 (L-C Kappe, R. F. Morse, and me as editors), AMS 2008; the paper is A classification of certain maximal subgroups of alternating groups, pp. 21-26, MR2478411.
As pointed out by Jack, this does exhaust all possible embeddings of $S_n$ into $S_k$ (presumably you are okay with the maps that are not embeddings...)
