  We develop algebro-combinatorial tools for computing the Thom polynomials for
the Morin singularities $A_i(-)$ ($i\ge 0$). The main tool is the function
$F^{(i)}_r$ defined as a combination of Schur functions with certain numerical
specializations of Schur polynomials as their coefficients. We show that the
Thom polynomial ${\cal T}^{A_i}$ for the singularity $A_i$ (any $i$) associated
with maps $({\bf C}^{\bullet},0) \to ({\bf C}^{\bullet+k},0)$, with any
parameter $k\ge 0$, under the assumption that $\Sigma^j=\emptyset$ for all
$j\ge 2$, is given by $F^{(i)}_{k+1}$. Equivalently, this says that "the
1-part" of ${\cal T}^{A_i}$ equals $F^{(i)}_{k+1}$. We investigate 2 examples
when ${\cal T}^{A_i}$ 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 Rim\'anyi et al. with the techniques of Schur functions.
