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The $m$-quasiinvariants of the symmetric group (denoted $QI_m$ for $m$ a nonnegative integer) were defined in 2002 by Feigin and Veselov. They form a nested sequence of $S_n$ modules with $QI_0$ being the polynomial ring in $n$ variables and $QI_\infinity$ being the symmetric functions (i.e., the invariants of $S_n$). Remarkably, they showed that for any $m$, the quotient $QI_m / QI_\infinity$ is isomorphic to the left regular representation of $S_n$. This suggests that there should be some nice combinatorics in the picture, and I will talk about some initial results in this direction. This is joint work with Gregg Musiker.