The M\"obius function of the consecutive pattern poset

TL;DR

This paper computes the M"obius function for intervals in the poset of permutations ordered by consecutive pattern containment, providing a complete solution and efficient algorithms for most cases.

Contribution

It offers a complete characterization and polynomial time algorithms for computing the M"obius function in the consecutive pattern poset.

Findings

M"obius function values are only -1, 0, or 1.

Provides a polynomial time algorithm for most intervals.

Achieves a complete solution to the M"obius function computation problem.

Abstract

An occurrence of a consecutive permutation pattern $p$ in a permutation $\pi$ is a segment of consecutive letters of $\pi$ whose values appear in the same order of size as the letters in $p$. The set of all permutations forms a poset with respect to such pattern containment. We compute the M\"obius function of intervals in this poset, providing what may be called a complete solution to the problem. For most intervals our results give an immediate answer to the question. In the remaining cases, we give a polynomial time algorithm to compute the M\"obius function. In particular, we show that the M\"obius function only takes the values -1, 0 and 1.