We show that if a permutation π contains two intervals of length 2, where one interval is an ascent and the other a descent, then the Möbius function µ[1, π] of the interval [1, π] is zero. As a consequence, we show 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.We also show that if a permutation φ can be expressed as a direct sum of the form α⊕1⊕β, then any permutation π containing an interval orderisomorphic to φ has µ[1, π] = 0; we deduce this from a more general result showing that µ[σ, π] = 0 whenever π contains an interval of a certain form. Finally, we show that if a permutation π contains intervals isomorphic to certain pairs of permutations, or to certain permutations of length six, then µ[1, π] = 0. * V. Jelínek and J.

We show that the Möbius function of an interval in a permutation poset where the lower bound is sum (resp. skew) indecomposable depends solely on the sum (resp. skew) indecomposable permutations contained in the upper bound, and that this can simplify the calculation of the Möbius sum. For increasing oscillations, we give a recursion for the Möbius sum which only involves evaluating simple inequalities.Our motivation for this paper is to find a contributing set C σ,π that is significantly smaller than the poset interval [σ, π), and a {0, ±1} weighting function(2)Plainly, in Equation 2, we could set C σ,π = [σ, π), and W (σ, α, π) = 1, which is equivalent to Equation 1.One approach here would be to take a permutation β such that σ < β < π. We could then set C σ,π = {λ : λ ∈ [σ, π) and λ ∈ [σ, β]}, and W (σ, α, π) = 1, since, from Equation 2, λ∈[σ,β] µ[σ, λ] = 0. This approach was used in Smith [5], who determined the Möbius function on the interval [1, π] for all permutations π with a single descent. Smith's paper is unusual, in that it provides an explicit formula for the value of the Möbius function.Our approach is different. We identify individual elements (say λ), of the poset that have µ[σ, λ] = 0. We also show that there are pairs of elements,, and so we can exclude these pairs of elements. Finally, we show that there are quartets of permutations λ 1 , . . . , λ 4 where 4 i=1 µ[σ, λ i ] = 0; and that we can systematically identify these quartets. By excluding these permutations from C σ,π we can significantly reduce the number of elements in C σ,π compared to the number of elements in the interval [σ, π). This approach results in the ability to compute µ[σ, π], where σ is indecomposable, much faster than evaluating Equation 1. For increasing oscillations, we will show that the elements of C σ,π can be determined using simple inequalities, and that as a consequence µ[σ, π] can be determined using inequalities. With this approach, we have computed µ [1, π], where π is an increasing oscillation, up to |π| = 2,000,000.The study of the Möbius function in the permutation poset was introduced by Wilf [9]. The first result in this area was by Sagan and Vatter [4], who determined the Möbius function on intervals of layered permutations. Steingrímsson and Tenner [8] found a large class of pairs of permutations (σ, π) where µ[σ, π] = 0, as well as determining the Möbius function where σ occurs exactly once in π, and σ and π satisfy certain other conditions. Burstein, Jelínek, Jelínková and Steingrímsson [1] found a recursion for the Möbius function for sum/skew decomposable permutations in terms of the sum/skew indecomposable permutations in the lower and upper bounds. They also found a method to determine the Möbius function for separable permutations by counting embeddings. We use the recursions for decomposable permutations to underpin the first part of this paper. McNamara and Steingrímsson [3] investigated the topology of intervals in the permutation poset, and found a single recurrence equivalent to the recursio...

We 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 define a method of constructing a permutation from a smaller permutation which we call "ballooning". We show that if β is a 2413balloon, and π is the 2413-balloon of β, then µ[1, π] = 2µ [1, β]. This allows us to construct a sequence of permutations π1, π2, π3 . . . with lengths n, n+ 4, n + 8, . . . such that µ[1, πi+1] = 2µ [1, πi], and this gives us exponential growth. Further, our construction method gives permutations that lie within a hereditary class with finitely many simple permutations.We also find an expression for the value of µ [1, π], where π is a 2413balloon, with no restriction on the permutation being ballooned.

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