  This paper contains an extensive combinatorial analysis of the single-peaked
domain restriction and investigates the likelihood that an election is
single-peaked. We provide a very general upper bound result for domain
restrictions that can be defined by certain forbidden configurations. This
upper bound implies that many domain restrictions (including the single-peaked
restriction) are very unlikely to appear in a random election chosen according
to the Impartial Culture assumption. For single-peaked elections, this upper
bound can be refined and complemented by a lower bound that is asymptotically
tight. In addition, we provide exact results for elections with few voters or
candidates. Moreover, we consider the P\'{o}lya urn model and the Mallows model
and obtain lower bounds showing that single-peakedness is considerably more
likely to appear for certain parameterizations.
