Counting linear extensions Graham Brightwell Peter Winkler Article Received: 06 June 1991 Accepted: 25 October 1991
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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Authors and Affiliations Graham Brightwell Peter Winkler 1. London School of Economics and Political Science London UK 2. Bellcore Morristown USA