Thom polynomials and schur functions:: Towards the singularities Ai(-)

We develop algebro-combinatorial tools for computing the Thom polynomials for the Morin singularities A(i)(-) (i >= 0). The main tool is the function F-r((i)) defined as a combination of Schur functions with certain numerical specializations of Schur polynomials as their coefficients. We show that the Thom polynomial T-Ai for the singularity A(i) (any i) associated with maps (C degrees, 0) -> (C degrees(+k), 0), with any parameter k >= 0, under the assumption that Sigma(j) = empty set for all j >= 2, is given by F-k+1((i)). Equivalently, this says that "the 1-part" of T-Ai equals F-k+1((i)). We investigate 2 examples when T-Ai apart from its 1-part consists also of the 2-part being a single Schur function with some multiplicity. Our computations combine the characterization of Thom polynomials via the "method of restriction equations" of Rimanyi et al. with the techniques of Schur functions.