Restricted Partition Posets and Specht Modules

PARTITIONS INTO EVEN AND ODD BLOCK SIZE AND SOME UNUSUAL CHARACTERS OF THE SYMMETRIC GROUPS - A. R. CALDERBANK, P. HANLON, and R. W. ROBINSON

http://plms.oxfordjournals.org/content/s3-53/2/288.full.pdf - the top homology group of the order complex of $\Pi_n^d\setminus \{\hat{1}\}$ is the Specht module on the border strip correspoding to the composition $(d, \ldots, d, d-1)$.

$\Pi_n^d$	$\tilde{H}(\Pi_n^d)$	dimension	Mobius function
$\Pi^\bullet_{\vec{c}}$	$\tilde{H}(\Pi^\bullet_{\vec{c}})$	dimension	Mobius function

Question

$\tilde{H}_{n-k-1}(\Pi_{(k,1,\ldots,1)}^\bullet \setminus \{\hat{1}\}) \cong_{S_n} \tilde{H}_{n-k-1}(B_n^k \setminus \{\hat{1}\}) \cong_{S_n} \text{Specht module of the border strip } (k,1,\ldots,1).$ Why?
$\Pi_{(k,1,\ldots,1)}^\bullet$ is the set of pointed partitions $\{B_1, B_2, \ldots, B_t, \underline{Z}\}$ such that at least one of $B_i$ has cardinality greater than or equal to $k$. On the other hand, $B_n^k = \{A \in B_n ~\vert ~ \lvert A\rvert \geq k\}$.
- Answer: $\tilde{H}_{n-k-1}(\Delta(\Pi_{(k,1,\ldots,1)}^\bullet \setminus \{\hat{1}\})) \cong \tilde{H}_{n-k-1} (\Delta_{(k,1,\ldots,1)}\setminus \{\hat{1}\}) \cong \tilde{H}_{n-k-1} (B_n(S)\setminus \{\hat{1}\}) \cong \tilde{H}_{n-k-1}(B_n^k \setminus \{\hat{1}\})$, where $S = \{k, k+1, \ldots, n\}$

Wach's EL-labeling of $d$-divisible partition lattice does not work in the case of pointed partition poset because some the rising chain in an interval might be missing in the pointed partition poset.