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The topology of the permutation pattern poset
, 2014
"... The set of all permutations, ordered by pattern containment, forms a poset. This extended abstract presents the first explicit major results on the topology of intervals in this poset. We show that almost all (open) intervals in this poset have a disconnected subinterval and are thus not shellable ..."
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The set of all permutations, ordered by pattern containment, forms a poset. This extended abstract presents the first explicit major results on the topology of intervals in this poset. We show that almost all (open) intervals in this poset have a disconnected subinterval and are thus not shellable. Nevertheless, there seem to be large classes of intervals that are shellable and thus have the homotopy type of a wedge of spheres. We prove this to be the case for all intervals of layered permutations that have no disconnected subintervals of rank 3 or more. We also characterize in a simple way those intervals of layered permutations that are disconnected. These results carry over to the poset of generalized subword order when the ordering on the underlying alphabet is a rooted forest. We conjecture that the same applies to intervals of separable permutations, that is, that such an interval is shellable if and only if it has no disconnected subinterval of rank 3 or more. We also present a simplified version of the recursive formula for the Möbius function of decomposable permutations given by Burstein et al.
Permutation Codes Correcting a Single Burst Deletion I: Unstable Deletions
"... Abstract—We construct the first class of permutation codes that are capable of correcting a burst of up to s unstable deletions, for general s. Efficient decoding algorithms are presented to show the correctness of our constructions. I. ..."
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Abstract—We construct the first class of permutation codes that are capable of correcting a burst of up to s unstable deletions, for general s. Efficient decoding algorithms are presented to show the correctness of our constructions. I.