There exists a unique map from permutations of 1,2,3... that move finitely many numbers, to their "Schubert polynomials" in ${\mathbb Z}[x_1,x_2,...]$, satisfying the following recursion: $S_{id} = 1$, and if $w(i) > w(i+1)$, then $S_{w r_i} = (S_w - r_i \cdot S_w) / (x_i - x_{i+1})$. (Here $r_i$ switches $i$ and $i+1$, or $x_i$ and $x_{i+1}$.)

It's not too hard to prove that these are polynomials, form a basis of the polynomial ring, have positive coefficients, and much else. The nonobvious theorem is that the structure constants (expanding a product of two basis elements in the basis) are positive. The only proofs known of this are geometric.

(This is perhaps a lame example, in that the motivation for Schubert polynomials was geometric -- they represent the classes of Schubert varieties in the cohomology rings of flag manifolds.)