Some irreducibility results for truncated binomial expansions

For positive integers n>k, let be the polynomial obtained by truncating the binomial expansion of n(1+x) at the kth stage. These polynomials arose in the investigation of Schubert calculus in Grassmannians. In this paper, the authors prove the irreducibility of Pn,k(x) over the field of rational numbers when 2≤2k≤n<3(k+1).