Positroids, non-crossing partitions, and positively oriented matroids

Federico Ardila, Felipe Rincón, Lauren Williams
2014 Discrete Mathematics & Theoretical Computer Science  
International audience 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
more » ... ^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.
doi:10.46298/dmtcs.2431 fatcat:vklnn5rudrbu5nse2ajo6mizjq