2413-balloon permutations and the growth of the Möbius function

Marchant, David William (2020). 2413-balloon permutations and the growth of the Möbius function. Electronic Journal of Combinatorics, 27(1), article no. P1.7.

DOI: https://doi.org/10.37236/8554

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 β is a 2413-balloon, and π is the 2413-balloon of β, then μ[1, π] = 2μ[1, β]. This allows us to construct a sequence of per-mutations π₁, π₂, π₃ ... with lengths n, n+4, n+8, ... such that μ[1, π_i+1] = 2μ[1, π_i], and this gives us exponential growth. Further, our construction method gives per-mutations 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 2413-balloon, with no restriction on the permutation being ballooned.