Birational sequences for the Grassmannian Gr(3,n)

Following the ideas of Bossinger and Fang, Fourier, and Littelman, we study iterated sequences for the Grassmannian $\mathrm{Gr}(3,n)$ as a special class of birational sequences. For each iterated sequence $S$, there is a weighting matrix $M_S$ corresponding to a valuation on the rational coordinate ring and we show that the initial form of a Plücker relation ${\mathrm{in}}_{M_S}(R_{I,J})$ is binomial. We show that, in some cases, the cones $C_S$ in the tropical Grassmannian that satisfy ${\mathrm{in}}_{M_S}({\mathcal{I}}_{3,n})={\mathrm{in}}_{C_S}({\mathcal{I}}_{3,n})$ only depend on the first two indices used in each iteration. In the case of $\mathrm{Gr}(3,6)$, these cones are obtained computationally and are classified up to automorphism induced by the symmetric group $S_6$.