The Rank Function of a Positroid and Non-Crossing Partitions
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A positroid is a special case of a realizable matroid that arose from the study of totally nonnegative part of the Grassmannian by Postnikov . Postnikov demonstrated that positroids are in bijection with certain interesting classes of combinatorial objects, such as Grassmann necklaces and decorated permutations. The bases of a positroid can be described directly in terms of the Grassmann necklace and decorated permutation . In this paper, we show that the rank of an arbitrary set in a positroid can be computed directly from the associated decorated permutation using non-crossing partitions.