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Séminaire Lotharingien de Combinatoire, 78B.23 (2017), 12 pp.

# Richard Ehrenborg and Dustin Hedmark

# Filters in the Partition Lattice

**Abstract.**
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.

Received: November 14, 2016.
Accepted: February 17, 2017.
Final version: April 1, 2017.

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