On counting and constructing all admissible orders of the non-empty intervals in any finite chain

Abstract

In many fields, there is a need to process data that only includes a finite number of values. To express the uncertainty regarding these values, one can use non-empty intervals, called epistemic, that are usually ordered in terms of the product, aka marginal, order. However, a partial order of this form is often insufficient in applications such as decision making, optimization, image segmentation, and edge detection. To this end, the given partial order of the non-empty intervals in a finite chain must be extended to a linear order, known as an admissible order. In this paper, we determine the number of all of these linear extensions and present an algorithm for generating them.