Counting linear extensions
 Graham Brightwell,
 Peter Winkler
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Abstract
We survey the problem of counting the number of linear extensions of a partially ordered set. We show that this problem is #Pcomplete, settling a longstanding open question. This result is contrasted with recent work giving randomized polynomialtime 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 #Phard. 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 #Pcomplete.
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 Title
 Counting linear extensions
 Journal

Order
Volume 8, Issue 3 , pp 225242
 Cover Date
 19910901
 DOI
 10.1007/BF00383444
 Print ISSN
 01678094
 Online ISSN
 15729273
 Publisher
 Kluwer Academic Publishers
 Additional Links
 Topics
 Keywords

 06A06
 68C25
 Partial order
 linear extension
 #Pcomplete
 Industry Sectors
 Authors

 Graham Brightwell ^{(1)}
 Peter Winkler ^{(2)}
 Author Affiliations

 1. London School of Economics and Political Science, Houghton Street, London, UK
 2. Bellcore, 445 South St., Morristown, New Jersey, USA