## 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## Preview

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## Copyright information

© Kluwer Academic Publishers 1991