# On The Möbius Function Of Permutations Under The Pattern Containment Order

Marchant, David William (2020). On The Möbius Function Of Permutations Under The Pattern Containment Order. PhD thesis The Open University.

## Abstract

We study several aspects of the Möbius function, μ[σ, π], on the poset of permutations under the pattern containment order.

First, we consider cases where the lower bound of the poset is indecomposable. We show that μ[σ, π] can be computed by considering just the indecomposable permutations contained in the upper bound. We apply this to the case where the upper bound is an increasing oscillation, and give a method for computing the value of the Möbius function that only involves evaluating simple inequalities.

We then consider conditions on an interval which guarantee that the value of the Möbius function is zero. In particular, we show that if a permutation π contains two intervals of length 2, which are not order-isomorphic to one another, then μ[1, π] = 0. This allows us to prove that the proportion of permutations of length n with principal Möbius function equal to zero is asymptotically bounded below by (1−1/e) 2 ≥ 0.3995. This is the first result determining the value of μ[1, π] for an asymptotically positive proportion of permutations π.

Following this, we use “2413-balloon” permutations to show that the growth of the principal Möbius function on the permutation poset is exponential. This improves on previous work, which has shown that the growth is at least polynomial.

We then generalise 2413-balloon permutations, and find a recursion for the value of the principal Möbius function of these generalisations.

Finally, we look back at the results found, and discuss ways to relate the results from each chapter. We then consider further research avenues.

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