, Volume 8, Issue 3, pp 225–242

Counting linear extensions


  • Graham Brightwell
    • London School of Economics and Political Science
  • Peter Winkler
    • Bellcore

DOI: 10.1007/BF00383444

Cite this article as:
Brightwell, G. & Winkler, P. Order (1991) 8: 225. doi:10.1007/BF00383444


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 height of an element x of a give poset, and of determining the probability that x lies below y in a random linear extension, are #P-complete.

AMS subject classifications (1991)


Key words

Partial orderlinear extension#P-complete
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Copyright information

© Kluwer Academic Publishers 1991