2413-Balloon Permutations and the Growth of the Möbius Function

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.