Counting linear extensions
Received: 06 June 1991 Accepted: 25 October 1991 DOI:
Cite this article as: Brightwell, G. & Winkler, P. Order (1991) 8: 225. doi:10.1007/BF00383444 Abstract
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
Research carried out while this author was visiting Bellcore under the auspices of DIMACS.
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