I will describe the combinatorics of Schubert curves, which are one-dimensional Schubert problems defined with respect to flags osculating the rational normal curve. The real geometry of such curves is described by orbits of a map \omega on skew tableaux, defined as the commutator of jeu de taquin rectification and promotion. In particular, the real locus of the Schubert curve naturally covers RP^1, with \omega as the monodromy operator.

I will give a local, faster algorithm for computing \omega without rectifying the tableau. Certain steps in the algorithm are in bijection with Pechenik and Yong's 'genomic tableaux', which enumerate the K-theoretic Littlewood-Richardson coefficient of the Schubert curve. As a corollary, I'll give purely combinatorial proofs of some numerical and geometric results relating the K-theory and real geometry of the curve. This is joint work with Maria Gillespie.