Partial order relation on subsets

This is the Bruhat order 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 $b_1 < b_2 < \cdots < b_{n-k}$.

This order is sometimes denoted $L(k,n-k)$ and has many interesting properties. See for instance Chapter 6 of http://math.mit.edu/~rstan/algcomb.pdf. For the characteristic polynomial of the Hasse diagram of $L(k,n-k)$ (considered as a graph), see Remark 5.6 of http://math.mit.edu/~rstan/papers/vac.pdf.

I call this the compression order on $[n]^{(r)}$, because it is the order induced by the operation of left-compression (or left-shifting) used in extremal set theory.