Sign representation of single-peaked preferences and Bruhat orders

TL;DR

This paper introduces a sign representation for single-peaked preferences on a social axis, enabling easier computation of domain sizes and revealing connections with Bruhat orders and rhombus tilings.

Contribution

It presents a novel sign representation for single-peaked preferences, linking them to Bruhat orders and rhombus tilings, and provides methods to compute domain cardinalities.

Findings

Sign representation simplifies domain size calculations.

Operations on the sign representation define the Bruhat poset.

Connections established with rhombus tiling and known results.

Abstract

Single-peaked preferences and domains are extensively researched in social science and economics. In this study, we examine the interval property as well as combinatorial structure of single-peaked preferences on a fixed Left-Right social axis. We introduce a sign representation of single-peaked preferences; consequently, some cardinalities of single-peaked domains are easily obtained. Basic operations on the sign representation, which completely define the Bruhat poset, are also provided. The applications to known results and an isomorphic relation with associated rhombus tiling are given. Finally, we some discussions of related topics.