Linear decision trees: volume estimates and topological bounds
STOC '92 Proceedings of the twenty-fourth annual ACM symposium on Theory of computing
Group representations on the homology of products of posets
Journal of Combinatorial Theory Series A
Partitions with Restricted Block Sizes, Möbius Functions, and the k-of-Each Problem
SIAM Journal on Discrete Mathematics
Modified Stirling Numbers and p-Divisibility in the Universal Typical pk-Series
Journal of Algebraic Combinatorics: An International Journal
Journal of Combinatorial Theory Series A
Exponential Dowling structures
European Journal of Combinatorics
Obstructions to shellability, partitionability, and sequential Cohen-Macaulayness
Journal of Combinatorial Theory Series A
Unimodality of Eulerian quasisymmetric functions
Journal of Combinatorial Theory Series A
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We generalize results of Calderbank, Hanlonand Robinson on the representation of the symmetric groupon the homology of posets of partitions with restricted block size. Calderbank, Hanlon and Robinson consider the cases of blocksizes that are congruent to 0 mod d and 1 mod d for fixed d. We derive a general formulafor the representation of the symmetric group on the homology ofposets of partitions whose block sizes are congruent tok mod d for any k and d. This formula reduces to the Calderbank-Hanlon-Robinson formulas when k = 0, 1 and to formulas of Sundaram for the virtual representation on the alternating sum ofhomology. Our results apply to restricted block size partition posetseven more general than the k mod d partition posets. These posets include the lattice ofpartitions whose block sizes are bounded from below by some fixed k. Our main tools involve the new theory of nonpure shellabilitydeveloped by Björner and Wachs and a generalization of a techniqueof Sundaram which uses Whitney homology to compute homologyrepresentations ofCohen-Macaulay posets. An application to subspace arrangementsis also discussed.