Bret Benesh and Ben Newton determined all pairs (m,n) such that Sm contains a maximal subgroup isomorphic to Sn. They are either (n+1,n) with the obvious inclusion (or mapping S5 into the image of a point stabilizer under the outer automorphism of S6); (binomnk,n), coming from the action of Sn on the subsets of k elements of 1,2,ldots,n; and ((kr)!/(r!)kk!,kr) with 1ltk,r, with Skr acting on the the right cosets of a maximal subgroups of the wreath product SkwrSr. 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 Sm has a maximal subgroup isomorphic to An; such that Am has a maximal subgroup isomorphic to Sn; and such that Am has a maximal subgroup isomorphic to An. 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 Sn into Sk (presumably you are okay with the maps that are not embeddings...)
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