The Mobius Function of the Permutation Pattern Poset
Abstract
A permutation τ contains another permutation σ as a pattern if τ has 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 σ occurs precisely once in τ, and σ and τ satisfy 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 σ in τ. We also conjecture that the Mobius function of the interval [1,τ] is -1, 0 or 1.