I consider all $k$ element subsets of the set $\{1,\ldots,n\}$ and define a partial order relation $\prec$ as follows: $\{a\_1,\ldots,a\_k\}\prec\{b\_1,\ldots,b\_k\}$, if and only if $a\_1<\cdots<a\_k$, $b\_1<\cdots<b\_k$ and $a\_i\leq b\_i$ for all $i=1,\ldots,k$.


What is this partial order commonly called in the literature? Can someone point out references?


Thanks a lot,
Torsten


[edit] Here is a Hasse diagram of this partial order for $n=6$ and $k=3$:
<http://www.freeimagehosting.net/7pmk7>



Partial order relation on subsets
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>.


