The Mobius Function of the Permutation Pattern Poset

TL;DR

This paper investigates the Mobius function within the permutation pattern poset, providing conditions for when it equals zero, one, or negative one, and proposing conjectures about its bounds in pattern-avoiding permutations.

Contribution

It characterizes the Mobius function for specific permutation intervals and introduces conjectures on its bounds in pattern-avoiding classes.

Findings

Identified classes of permutation pairs with Mobius function zero.

Solved the Mobius function for permutations with a unique occurrence of a pattern.

Conjectured bounds for the Mobius function in pattern-avoiding permutation intervals.

Abstract

A permutation \tau contains another permutation \sigma as a pattern if \tau has a subsequence whose elements are in the same order with respect to size as the elements in \sigma. This defines a partial order on the set of all permutations, and gives a graded poset P. We give a large class of pairs of permutations whose intervals in P have Mobius function 0. Also, we give a solution to the problem when \sigma occurs precisely once in \tau, and \sigma and \tau satisfy certain further conditions, in which case the Mobius function is shown to be either -1, 0 or 1. We conjecture that for intervals [\sigma,\tau] consisting of permutations avoiding the pattern 132, the magnitude of the Mobius function is bounded by the number of occurrences of \sigma in \tau. We also conjecture that the Mobius function of the interval [1,\tau] is -1, 0 or 1.