The Möbius Function of the Permutation Pattern Poset

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 Möbius 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 Möbius 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 Möbius function is bounded by the number of occurrences of σ in τ . We also conjecture that the Möbius function of the interval [1, τ ] is −1, 0 or 1.