NDLI: Combinatorial Markov chains on linear extensions

Content Provider	Springer Nature Link
Author	Ayyer, Arvind Klee, Steven Schilling, Anne
Copyright Year	2013
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.
Starting Page	853
Ending Page	881
Page Count	29
File Format	PDF
ISSN	09259899
Journal	Journal of Algebraic Combinatorics
Volume Number	39
Issue Number	4
e-ISSN	15729192
Language	English
Publisher	Springer US
Publisher Date	2013-09-10
Publisher Place	Boston
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Linear extensions Posets Promotion operator Markov chains Combinatorics Convex and Discrete Geometry Order, Lattices, Ordered Algebraic Structures Computer Science Group Theory and Generalizations
Content Type	Text
Resource Type	Article
Subject	Discrete Mathematics and Combinatorics Algebra and Number Theory

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