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.

翻译：我们研究了具有给定位置矩阵的选举的性质（在此类选举中，每位候选人按照矩阵指定的票数占据每个排名位置）。我们证明，统计生成给定位置矩阵的选举数量是#P完全的。因此，均匀随机抽样此类选举似乎具有挑战性，我们提出了一种更简单的算法，虽无严格保证。接下来，我们考虑检验给定矩阵能否通过具有特定结构（例如单峰性或群体可分离性）的选举实现的问题。最后，我们考虑检验给定位置矩阵能否通过具有孔多塞赢家的选举实现的问题。我们通过实验补充了理论发现。