Discrete CATS Seminar
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U N I V E R S I T Y
O F
K E N T U C K Y

DISCRETE
CATS
SEMINAR

WHERE CATS =
COMBINATORICS,
ALGEBRA,
TOPOLOGY
&
STATISTICS!

845 PATTERSON OFFICE TOWER

2011 - 2012

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DOCTORAL DEFENSE
"Analytic and topological combinatorics of
partition posets and permutations"

JiYoon Jung

University of Kentucky

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Thursday, April 19, 2012

10:00 am

Location TBA

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Abstract:

For each composition c we show that the order complex of the poset of
pointed set partitions is a wedge of spheres of the same dimensions
with the multiplicity given by the number of permutations with descent
composition c. Furthermore, the action of the symmetric group on the
top homology is isomorphic to the Specht module of a border strip
associated to the composition. 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.

Next, we extend the notion of consecutive pattern avoidance to
considering sums over all permutations where each term is a product of
weights depending on each consecutive pattern of a fixed length. We
study the problem of finding the asymptotics of these sums. Our
technique is to extend the spectral method of Ehrenborg, Kitaev and
Perry. When the weight depends on the descent pattern we show how to
find the equation determining the spectrum. We give two length 4
applications. First, we find the asymptotics of the number of
permutations with no triple ascents and no triple descents. Second we
give the asymptotics of the number of permutations with no isolated
ascents or descents. Our next result is a weighted pattern of length 3
where the associated operator only has one non-zero eigenvalue. Using
generating functions we show that the error term in the asymptotic
expression is the smallest possible.