  We present an LLT-type formula for a general power of the nabla operator
applied to the Cauchy product for the modified Macdonald polynomials, and use
it to deduce a new proof of the generalized shuffle theorem describing
$\nabla^k e_n$, and the Elias-Hogancamp formula for $(\nabla^k p_1^n,e_n)$ as
corollaries. We give a direct proof of the theorem by verifying that the LLT
expansion satisfies the defining properties of $\nabla^k$, such as
triangularity in the dominance order, as well as a geometric proof based on a
method for counting bundles on $\mathbb{P}^1$ due to the second author. These
formulas are related to an affine paving of the type A unramified affine
Springer fiber studied by Goresky, Kottwitz, and MacPherson, and also to
Stanley's chromatic symmetric functions.
