  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.

Combinatorics;Computer Science and Game Theory