  Gaussian elimination with partial pivoting (GEPP) is a widely used method to
solve dense linear systems. Each GEPP step uses a row transposition pivot
movement if needed to ensure the leading pivot entry is maximal in magnitude
for the leading column of the remaining untriangularized subsystem. We will use
theoretical and numerical approaches to study how often this pivot movement is
needed. We provide full distributional descriptions for the number of pivot
movements needed using GEPP using particular Haar random ensembles, as well as
compare these models to other common transformations from randomized numerical
linear algebra. Additionally, we introduce new random ensembles with fixed
pivot movement counts and fixed sparsity, $\alpha$. Experiments estimating the
empirical spectral density (ESD) of these random ensembles leads to a new
conjecture on a universality class of random matrices with fixed sparsity whose
scaled ESD converges to a measure on the complex unit disk that depends on
$\alpha$ and is an interpolation of the uniform measure on the unit disk and
the Dirac measure at the origin.
