Abstract
The poset P of all permutations ordered by pattern containment is a fundamental object of study in the field of permutation patterns. This poset has a very rich and complex topology and an understanding of its Möbius function has proved particularly elusive, although results have been slowly emerging in the last few years. Using a variety of topological techniques we present a two term formula for the Mo ̈bius function of intervals in P. The first term in this formula is, up to sign, the number of so called normal occurrences of one permutation in another. Our definition of normal occurrences is similar to those that have appeared in several variations in the literature on the Möbius function of this and other posets, but simpler than most of them. The second term in the formula is (still) complicated, but we conjecture that it equals zero for a significant proportion of intervals. We present some cases where the second term vanishes and others where it is nonzero. Computing the Möbius function recursively from its definition has exponential complexity, whereas the computation of the first term in our formula is polynomial and the exponential part is isolated to the second term, which seems to often vanish. This is thus the first polynomial time formula for the Möbius function of what appears to be a large proportion of all intervals of P.
Highlights
Let σ and π be permutations of positive integers
The permutation poset P consists of all permutations with the partial order σ ≤ π if there is an occurrence of σ in π
We present a formula, in Theorem 21, that shows the Mobius function of [σ, π] is, up to a sign, equal to the number of normal occurrences of σ in π plus an extra term that seems to vanish for a significant proportion of intervals
Summary
Let σ and π be permutations of positive integers. We define an occurrence of σ in π to be a subsequence of π with the same relative order of size as the letters in σ. We present a formula, in Theorem 21, that shows the Mobius function of [σ, π] is, up to a sign, equal to the number of normal occurrences of σ in π plus an extra term that seems to vanish for a significant proportion of intervals. We show that the value of the second term of our formula for the Mobius function of [σ, π] is often nonzero when π has a decomposition into a direct sum with consecutive equal components Such repeated components have played a (different) role in work by other authors, but the connection remains mysterious
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