DOCTORAL DEFENSE

"Analytic and topological combinatorics of partition posets and permutations"

JiYoon Jung
University of Kentucky

Thursday, April 19, 2012
10:00 am
Location TBA

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.