Abstract
We show that the growth of the principal Möbius function on the permutation poset is at least 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 $\beta$ is a 2413-balloon, and $\pi$ is the 2413-balloon of $\beta$, then $\mu[1,\pi] = 2 \mu[1,\beta]$. This allows us to construct a sequence of permutations $\pi_1, \pi_2, \pi_3\ldots$ with lengths $n, n+4, n+8, \ldots$ such that $\mu[1,\pi_{i+1}] = 2 \mu[1,\pi_{i}]$, 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 $\mu[1,\pi]$, where $\pi$ is a 2413-balloon, with no restriction on the permutation being ballooned.
Highlights
Let σ and π be permutations of natural numbers, written in one-line notation, with σ = σ1σ2 . . . σm, and π = π1π2 . . . πn
For ΦB(c) we show that some weaker conditions hold for an arbitrary subset of the reductions of π, and when we have an explicit set of proper reductions, we show that all conditions hold
Once we have shown that we have parity-reversing involutions, we will show how to express the Hall sum of R in terms of μ[β]
Summary
If β = 25314 (which is a 2413-balloon), we can construct a hereditary class that contains only the simple permutations {1, 12, 21, 2413, 25314}, where the growth of the principal Mobius function is exponential, answering questions in Burstein et al [4] and Jelınek et al [5]. A set of permutations S where every σ ∈ S satisfies 1 < σ < π, if C is the set of chains in the poset interval [1, π] where the second-highest element is in S, the Hall sum of C is − σ∈S μ[σ]. Since the chains in R are defined by the secondhighest element (κc) being in Rπ, the final part of the observation follows by applying Corollary 7
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