# 2413-balloon permutations and the growth of the M\"obius function

## TL;DR

This paper demonstrates that the principal M"obius function on the permutation poset grows exponentially, introducing a 'ballooning' construction method that systematically produces permutations with exponentially increasing M"obius values.

## Contribution

It introduces the 'ballooning' construction for permutations and proves it leads to exponential growth of the M"obius function, advancing understanding of permutation poset properties.

## Key findings

- Exponential growth of the M"obius function established
- 'Ballooning' method constructs permutations with predictable M"obius values
- Permutations lie within a hereditary class with finitely many simple permutations

## Abstract

We show that the growth of the principal M\"obius 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 $\beta$ 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.