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On the homology of partitions with an even number of blocks
Shelia Sundaram
Department of Mathematics & Computer Science
University of Miami
Coral Gables, FL
shelia@paris-gw.cs.miami.edu
The symmetric group acts naturally on the lattice of partitions of
a finite set, and this action extends to the homology of the
associated order complex. The same is true for the poset of
partitions with an even number of blocks. It is known that this
(rank-selected) subposet is homotopy equivalent to a wedge of spheres.
In this talk we discuss our recent work on the homology of partitions
of a set of cardinality $2n$ with an even number of blocks.
A preliminary analysis suggests that the action of the symmetric group
$S_{2n}$
does not permute the spheres in the associated order complex.
Nevertheless, we shall present overwhelming evidence
to support the conjecture that the representation
of $S_{2n}$ on the homology is in fact a permutation representation.
The idea of finding enumerative invariants of combinatorial interest
by considering group actions on various posets,
appears in a 1982 paper of Richard Stanley.
Our analysis of the homology representation has led to the discovery
of
some new enumerative invariants associated to the subposet
of this talk.
The representation-theoretic conjecture is equivalent to the
nonnegativity of these integers. These invariants
have interesting combinatorial connections with other well-known
families of numbers, most notably the tangent numbers.