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The topology of restricted partition posets

Abstract : For each composition $\vec{c}$ we show that the order complex of the poset of pointed set partitions $Π ^• _{\vec{c}}$ is a wedge of $β\vec{c}$ spheres of the same dimensions, where $β\vec{c}$ is 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 module $S^B$ where $B$ is a border strip associated to the composition $\vec{c}$. We also study the filter of pointed set partitions generated by a knapsack integer partitions and show the analogous results on homotopy type and action on the top homology.
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Richard Ehrenborg, Jiyoon Jung. The topology of restricted partition posets. 23rd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2011), 2011, Reykjavik, Iceland. pp.281-292, ⟨10.46298/dmtcs.2910⟩. ⟨hal-01215111⟩

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