  We show that if X_n is a variety of cxn-matrices that is stable under the
group Sym([n]) of column permutations and if forgetting the last column maps
X_n into X_{n-1}, then the number of Sym([n])-orbits on irreducible components
of X_n is a quasipolynomial in n for all sufficiently large n. To this end, we
introduce the category of affine FI^op-schemes of width one, review existing
literature on such schemes, and establish several new structural results about
them. In particular, we show that under a shift and a localisation, any
width-one FI^op-scheme becomes of product form, where X_n=Y^n for some scheme Y
in affine c-space. Furthermore, to any FI^op-scheme of width one we associate a
component functor from the category FI of finite sets with injections to the
category PF of finite sets with partially defined maps. We present a
combinatorial model for these functors and use this model to prove that
Sym([n])-orbits of components of X_n, for all n, correspond bijectively to
orbits of a groupoid acting on the integral points in certain rational
polyhedral cones. Using the orbit-counting lemma for groupoids and theorems on
quasipolynomiality of lattice point counts, this yields our Main Theorem.
