, Volume 8, Issue 3, pp 225-242

First online:

Counting linear extensions

  • Graham BrightwellAffiliated withLondon School of Economics and Political Science
  • , Peter WinklerAffiliated withBellcore

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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