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The Mobius Function of the Permutation Pattern Poset

A permutation τcontains another permutation σas a pattern if τhas a subsequence whose elements are in the same order with respect to size as the elements in σ. This defines a partial order on the set of all permutations, and gives a graded poset P. We give a large class of pairs of permutations whose intervals in P have Mobius function 0. Also, we give a solution to the problem when σoccurs precisely once in τ, and σand τsatisfy certain further conditions, in which case the Mobius function is shown to be either -1, 0 or 1. We conjecture that for intervals [σ,τ] consisting of permutations avoiding the pattern 132, the magnitude of the Mobius function is bounded by the number of occurrences of σin τ. We also conjecture that the Mobius function of the interval [1,τ] is -1, 0 or 1.