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 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 2413balloon, and π is the 2413-balloon of β, then µ[1, π] = 2µ [1, β]. This allows us to construct a sequence of permutations π1, π2, π3 . . . with lengths n, n+ 4, n + 8, . . . such that µ[1, πi+1] = 2µ [1, πi], 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 µ [1, π], where π is a 2413balloon, with no restriction on the permutation being ballooned.

On Permutations Avoiding Partially Ordered Patterns Defined by Bipartite Graphs

“…The recursion for b(n) is the same as that for c(n) in the proof of Theorem 10, so we can conclude from that proof that b(n) = a(n) − 2a(n − 1) + 1, which gives us the desired recursion for a(n) from (7).…”

“…Permutation patterns are an active area of research that attracts much attention in the literature (e.g. see [1,2,3,7] and references therein).…”

On the growth of the Möbius function of permutations

“…Very recently, as this paper was undergoing review, Marchant has published a preprint [10] with a construction of a sequence of permutation whose principal Möbius function is exponential in their size. This result, if confirmed, would greatly improve upon our polynomial lower bound on f (n), and also answer in the negative the first question in Problem 4.4, since Marchant's construction is based on permutations from a class with only finitely many simple permutations.…”