On the M\"{o}bius function of permutations under the pattern containment order

TL;DR

This paper investigates the M"{o}bius function in the permutation pattern poset, providing methods for computation, identifying conditions for zero values, and analyzing growth patterns, with implications for understanding permutation structures.

Contribution

It introduces new techniques for computing the M"{o}bius function, establishes conditions for zero values, and demonstrates exponential growth, advancing the understanding of permutation pattern posets.

Findings

The M"{o}bius function can be computed using indecomposable permutations.

Permutations with two distinct length-2 intervals have zero M"{o}bius function.

The growth of the principal M"{o}bius function is exponential.

Abstract

We study several aspects of the M\"{o}bius function, $\mu[\sigma,\pi]$, on the poset of permutations under the pattern containment order. First, we consider cases where the lower bound of the poset is indecomposable. We show that $\mu[\sigma,\pi]$ can be computed by considering just the indecomposable permutations contained in the upper bound. We apply this to the case where the upper bound is an increasing oscillation, and give a method for computing the value of the M\"{o}bius function that only involves evaluating simple inequalities. We then consider conditions on an interval which guarantee that the value of the M\"{o}bius function is zero. In particular, we show that if a permutation $\pi$ contains two intervals of length 2, which are not order-isomorphic to one another, then $\mu[1,\pi] = 0$. This allows us to prove that the proportion of permutations of length $n$ with principal M\"{o}bius function equal to zero is asymptotically bounded below by $(1-1/e)^2 \ge 0.3995$. This is the first result determining the value of $\mu[1,\pi]$ for an asymptotically positive proportion of permutations $\pi$. Following this, we use ''2413-balloon'' permutations to show that the growth of the principal M\"{o}bius function on the permutation poset is exponential. This improves on previous work, which has shown that the growth is at least polynomial. We then generalise 2413-balloon permutations, and find a recursion for the value of the principal M\"{o}bius function of these generalisations.