Geodesic
Positroids, non-crossing partitions, and positively oriented matroids
Discrete mathematics & theoretical computer science, DMTCS Proceedings vol. AT, 26th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2014), DMTCS Proceedings vol. AT, 26th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2014) (2014).

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We investigate the role that non-crossing partitions play in the study of positroids, a class of matroids introduced by Postnikov. We prove that every positroid can be constructed uniquely by choosing a non-crossing partition on the ground set, and then freely placing the structure of a connected positroid on each of the blocks of the partition. We use this to enumerate connected positroids, and we prove that the probability that a positroid on [n] is connected equals $1/e^2$ asymptotically. We also prove da Silva's 1987 conjecture that any positively oriented matroid is a positroid; that is, it can be realized by a set of vectors in a real vector space. It follows from this result that the positive matroid Grassmannian (or <i>positive MacPhersonian</i>) is homeomorphic to a closed ball. 
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     author = {Ardila, Federico and Rinc\'on, Felipe and Williams, Lauren},
     title = {Positroids, non-crossing partitions, and positively oriented matroids},
     journal = {Discrete mathematics & theoretical computer science},
     publisher = {mathdoc},
     volume = {DMTCS Proceedings vol. AT, 26th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2014)},
     year = {2014},
     doi = {10.46298/dmtcs.2431},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.2431/}
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Ardila, Federico; Rincón, Felipe; Williams, Lauren. Positroids, non-crossing partitions, and positively oriented matroids. Discrete mathematics & theoretical computer science, DMTCS Proceedings vol. AT, 26th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2014), DMTCS Proceedings vol. AT, 26th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2014) (2014). doi : 10.46298/dmtcs.2431. http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.2431/

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