This is the [Bruhat order][1] on $S_n / (S_k \times S_{n-k})$, which models the inclusion relations of Schubert varieties on the Grassmannian $Gr(k,n)$.  The elements of $S_n / (S_k \times S_{n-k})$ are generally modeled by their minimal length representatives, called Grassmann permutations.  You can biject your $k$-subset of $[n]$ to a Grassmann permutation by sending a $k$-subset $\lbrace\{ a_1, \dots, a_k \rbrace\}$ (with $a_1 < a_2 < \dots < a_k$) to the permutation of $[n]$ whose window (one-line) notation is $a_1 a_2 \cdots a_k b_1 b_2 \cdots b_{n-k}$ with $\lbrace\{ b_1, b_2, \dots, b_{n-k} \rbrace\}$ denoting the complement of your given set, again ordered in increasing order $b_1 < b_2 < \cdots < b_{n-k}$.


  [1]: http://en.wikipedia.org/wiki/Bruhat_order