The Möbius function of permutations with an indecomposable lower bound

Abstract

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...

On the growth of the Möbius function of permutations

“…This example, pointed out by Smith [15], gave the largest previously known growth of |µ(1, π)| in terms of |π|. We note that Brignall and Marchant [5] have recently found another, substantially different example of a family of permutations π for which they conjecture that |µ(1, π)| is quadratic in |π|.…”

2413-Balloon Permutations and the Growth of the Möbius Function

“…These are well-known consequences of Propositions 1 and 2 of Burstein, Jelínek, Jelínková and Steingrímsson [4], and we refrain from providing proofs here. The reader is directed to Lemma 4 in [3] for a proof of Lemma 1. Lemma 2 is a trivial extension of Corollary 3 in [4].…”

Intervals of permutations and the principal Möbius function

“…There are cases where we can determine that the principal Möbius function of a permutation is zero by examining part of the permutation, but the permutation is not strongly zero. As an example, if a permutation π, with |π| > 2, begins 12, then as a consequence of Propositions 1 and 2 in Burstein, Jelínek, Jelínková and Steingrímsson [5] (first stated explicitly as Lemma 4 in Brignall and Marchant [4]), µ[π] = 0. Such a permutation is not strongly zero, since 12 ∈ SZ.…”

“…Brignall and Marchant [4] showed that if the lower bound of an interval is indecomposable, then the Möbius function depends only on the indecomposable permutations contained in the upper bound, and used this result to find a fast polynomial algorithm for finding µ[π] where π is an increasing oscillation.…”