I need the following result for an example in a paper I'm writing. It's easy enough to prove, but I'd prefer to just give a reference. Does anyone know one?


Fix $1 \leq k \leq n$. Define $X\_{n,k}$ to be the following poset. The elements of $X\_{n,k}$ are ordered sequences $\omega = (x\_1,\ldots,x\_m)$, where the $x\_i$ are distinct elements of the $n$-element set $\{1,\ldots,n\}$ and $m \geq k$. The order relation is that $\omega\_1 \leq \omega\_2$ if $\omega\_1$ is a subsequence of $\omega\_2$. The theorem then is that the geometric realization of $X\_{n,k}$ is $(n-1-k)$-connected.



Posets of finite sequences are highly connected
If I understand your question correctly, an answer should appear in the paper "On lexicographically shellable posets" of Anders Bj\"orner and Michelle Wachs, in Transactions of the AMS 277, pp. 323-341. 


