Filters in the Partition Lattice

Richard Ehrenborg ; Dustin Hedmark

Séminaire lotharingien de combinatoire, 78B (2017)

Résumé

Given a filter Δ in the poset of compositions of n, we form the filter Π^*_Δ in the partition lattice. We determine all the reduced homology groups of the order complex of Π^*_Δ as S_n-1-modules in terms of the reduced homology groups of the simplicial complex Δ and in terms of Specht modules of border shapes. We also obtain the homotopy type of this order complex. These results generalize work of Calderbank--Hanlon--Robinson and Wachs on the d-divisible partition lattice. Our main theorem applies to a plethora of examples, including filters associated to integer knapsack partitions and filters generated by all partitions having block sizes a or b. We also obtain the reduced homology groups of the filter generated by all partitions having block sizes belonging to the arithmetic progression a, a+d, ..., a+(a-1)d, extending work of Browdy.