Journal of Algebraic Combinatorics

, Volume 39, Issue 4, pp 853–881

Combinatorial Markov chains on linear extensions

Article
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Abstract

We consider generalizations of Schützenberger’s promotion operator on the set \(\mathcal{L}\) of linear extensions of a finite poset of size n. This gives rise to a strongly connected graph on \(\mathcal{L}\). By assigning weights to the edges of the graph in two different ways, we study two Markov chains, both of which are irreducible. The stationary state of one gives rise to the uniform distribution, whereas the weights of the stationary state of the other have a nice product formula. This generalizes results by Hendricks on the Tsetlin library, which corresponds to the case when the poset is the anti-chain and hence \(\mathcal{L}=S_{n}\) is the full symmetric group. We also provide explicit eigenvalues of the transition matrix in general when the poset is a rooted forest. This is shown by proving that the associated monoid is \(\mathcal {R}\)-trivial and then using Steinberg’s extension of Brown’s theory for Markov chains on left regular bands to \(\mathcal {R}\)-trivial monoids.

Keywords

Linear extensions Posets Promotion operator Markov chains 

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department of MathematicsUC DavisDavisUSA
  2. 2.Department of MathematicsIndian Institute of ScienceBangaloreIndia
  3. 3.Department of MathematicsSeattle UniversitySeattleUSA

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