Counting Partial Orders with a Fixed Number of Comparable Pairs

Authors:
Hans Jürgen Prömel;Angelika Steger;Anusch Taraz
Affiliations:
Institut für Informatik, Humboldt–Universität zu Berlin, 10099 Berlin, Germany (e-mail: proemel@informatik.hu-berlin.de, taraz@informatik.hu-berlin.de);Institut für Informatik, Technische Universität München, 80290 München, Germany (e-mail: steger@in.tum.de);Institut für Informatik, Humboldt–Universität zu Berlin, 10099 Berlin, Germany (e-mail: proemel@informatik.hu-berlin.de, taraz@informatik.hu-berlin.de)
Venue:
Combinatorics, Probability and Computing
Year:
2001

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Abstract

In 1978, Dhar suggested a model of a lattice gas whose states are partial orders. In this context he raised the question of determining the number of partial orders with a fixed number of comparable pairs. Dhar conjectured that in order to find a good approximation to this number, it should suffice to enumerate families of layer posets. In this paper we prove this conjecture and thereby prepare the ground for a complete answer to the question.