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 $\operatorname{Gr} (3, n)$ 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 Pl\"{u}cker relation $\operatorname{in}_{M_S} (R_{I,J} )$ is binomial. We show that, in some cases, the cones $C_S$ in the tropical Grassmannian that satisfy $\operatorname{in}_{M_S} (\mathcal{I}_{3,n}) = \operatorname{in}_{C_S} (\mathcal{I}_{3,n})$ only depend on the first two indices used in each iteration. In the case of $\operatorname{Gr} (3, 6)$, these cones are obtained computationally and are classified up to automorphism induced by the symmetric group $S_6$.