Ranking patterns of unfolding models of codimension one

TL;DR

This paper investigates the enumeration of ranking patterns generated by unfolding models of codimension one, linking them to braid arrangements and chambers, and provides bounds on their counts.

Contribution

It introduces a novel geometric approach to count ranking patterns of unfolding models using braid arrangements and chambers.

Findings

Number of ranking patterns expressed via braid arrangements.

All braid slices correspond to chambers of an arrangement.

Provides an upper bound for ranking patterns ignoring object permutations.

Abstract

We consider the problem of counting the number of possible sets of rankings (called ranking patterns) generated by unfolding models of codimension one. We express the ranking patterns as slices of the braid arrangement and show that all braid slices, including those not associated with unfolding models, are in one-to-one correspondence with the chambers of an arrangement. By identifying those which are associated with unfolding models, we find the number of ranking patterns. We also give an upper bound for the number of ranking patterns when the difference by a permutation of objects is ignored.