We prove a bijection between the triangulations of the 3-dimensional cyclic poly-tope C(n + 2, 3) and persistent graphs with n vertices. We show that under this bijection the Stasheff-Tamari orders on triangulations naturally translate to subgraph inclusion between persistent graphs. Moreover, we describe a connection to the second higher Bruhat order B(n, 2). We also give an algorithm to efficiently enumerate all persistent graphs on n vertices and thus all triangulations of C(n + 2, 3).
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We thank Lito Goldmann for his work on enumerating persistent graphs, which led us to the discovery of the bijection. We further thank André Nichterlein for helpful initial discussions. We also thank the anonymous reviewers of Combinatorica for their valuable comments.
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Froese, V., Renken, M. Persistent Graphs and Cyclic Polytope Triangulations. Combinatorica 41, 407–423 (2021). https://doi.org/10.1007/s00493-020-4369-5
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