We describe T -equivariant Schubert calculus on G(k, n), T being an n-dimensional torus, through derivations on the exterior algebra of a free A-module of rank n, where A is the T -equivariant cohomology of a point. In particular, T -equivariant Pieri's formulas will be determined, answering a question raised by Lakshmibai, Raghavan and Sankaran (Equivariant Giambelli and determinantal restriction formulas for the Grassmannian, Pure Appl. Math. Quart. 2 (2006), 699-717).

We prove Newton's binomial formulas for Schubert Calculus to determine numbers of base point free linear series on the projective line with prescribed ramification divisor supported at given distinct points.Key words: Schubert Calculus on a Grassmann algebra, Newton's binomial formulas in Schubert calculus, enumerative geometry of linear series on the projective line.