  We study the space, $R_m$, of $m$-symmetric functions consisting of
polynomials that are symmetric in the variables $x_{m+1},x_{m+2},x_{m+3},\dots$
but have no special symmetry in the variables $x_1,\dots,x_m$. We obtain
$m$-symmetric Macdonald polynomials by $t$-symmetrizing non-symmetric Macdonald
polynomials, and show that they form a basis of $R_m$. We define $m$-symmetric
Schur functions through a somewhat complicated process involving their dual
basis, tableaux combinatorics, and the Hecke algebra generators, and then prove
some of their most elementary properties. We conjecture that the $m$-symmetric
Macdonald polynomials (suitably normalized and plethystically modified) expand
positively in terms of $m$-symmetric Schur functions. We obtain relations on
the $(q,t)$-Koska coefficients $K_{\Omega \Lambda}(q,t)$ in the $m$-symmetric
world, and show in particular that the usual $(q,t)$-Koska coefficients are
special cases of the $K_{\Omega \Lambda}(q,t)$'s. Finally, we show that when
$m$ is large, the positivity conjecture, modulo a certain subspace, becomes a
positivity conjecture on the expansion of non-symmetric Macdonald polynomials
in terms of non-symmetric Hall-Littlewood polynomials.
