, Volume 8, Issue 3, pp 225–242

Counting linear extensions

  • Graham Brightwell
  • Peter Winkler


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)

06A06 68C25 

Key words

Partial order linear extension #P-complete 


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Copyright information

© Kluwer Academic Publishers 1991

Authors and Affiliations

  • Graham Brightwell
    • 1
  • Peter Winkler
    • 2
  1. 1.London School of Economics and Political ScienceLondonUK
  2. 2.BellcoreMorristownUSA

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