Quasiinvariants of S3

Let sij represent a transposition in Sn. A polynomial P in Q[Xn] is said to be m-quasiinvariant with respect to Sn if (xi-xj)2m+1 divides (1-sij)P for all 1⩽i,j⩽n. We call the ring of m-quasiinvariants, QIm[Xn]. We describe a method for constructing a basis for the quotient QIm[X3]/(e1,e2,e3). This leads to the evaluation of certain binomial determinants that are interesting in their own right.