  We study the properties of elections that have a given position matrix (in
such elections each candidate is ranked on each position by a number of voters
specified in the matrix). We show that counting elections that generate a given
position matrix is #P-complete. Consequently, sampling such elections uniformly
at random seems challenging and we propose a simpler algorithm, without hard
guarantees. Next, we consider the problem of testing if a given matrix can be
implemented by an election with a certain structure (such as single-peakedness
or group-separability). Finally, we consider the problem of checking if a given
position matrix can be implemented by an election with a Condorcet winner. We
complement our theoretical findings with experiments.
