The topology of restricted partition posets
For each composition \(\vec {c}\) we show that the order complex of the poset of pointed set partitions\(\varPi^{\bullet}_{\vec {c}}\) is a wedge of spheres of the same dimension with the multiplicity given by the number of permutations with descent composition \(\vec {c}\). Furthermore, the action of the symmetric group on the top homology is isomorphic to the Specht moduleS B where B is a border strip associated to the composition. We also study the filter of pointed set partitions generated by a knapsack integer partition and show the analogous results on homotopy type and action on the top homology.
knapsack integer partitions border strip specht module complex of ordered set partitions descent set statistics poset of pointed set partitions