Counting linear extensions

@article{Brightwell1991CountingLE,
  title={Counting linear extensions},
  author={Graham R. Brightwell and Peter Winkler},
  journal={Order},
  year={1991},
  volume={8},
  pages={225-242}
}
We survey the problem of counting the number of linear extensions of a partially ordered set. We show that this problem is #P-complete, settling a long-standing open question. This result is contrasted with recent work giving randomized polynomial-time algorithms for estimating the number of linear extensions.One consequence of our main result is that computing the volume of a rational polyhedron is strongly #P-hard. We also show that the closely related problems of determining the average… 

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