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{\bf Matthew Macauley and Henning S. Mortveit }
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{\bf Posets from Admissible Coxeter Sequences}
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  We study the equivalence relation on the set of acyclic orientations
  of an undirected graph $\Gamma$ generated by source-to-sink
  conversions. These conversions arise in the contexts of admissible
  sequences in Coxeter theory, quiver representations, and
  asynchronous graph dynamical systems. To each equivalence class we
  associate a poset, characterize combinatorial properties of these
  posets, and in turn, the admissible sequences. This allows us to
  construct an explicit bijection from the equivalence classes over
  $\Gamma$ to those over $\Gamma'$ and $\Gamma''$, the graphs obtained
  from $\Gamma$ by edge deletion and edge contraction of a fixed
  cycle-edge, respectively.  This bijection yields quick and elegant
  proofs of two non-trivial results: $(i)$ A complete combinatorial
  invariant of the equivalence classes, and $(ii)$ a solution to the
  conjugacy problem of Coxeter elements for simply-laced Coxeter
  groups. The latter was recently proven by H.~Eriksson and
  K.~Eriksson using a much different approach.

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