Gaussian elimination with partial pivoting (GEPP) remains the most used dense linear solver. For a nxn matrix A, GEPP results in the factorization PA = LU where L and U are lower and upper triangular matrices and P is a permutation matrix. If A is a random matrix, then the associated permutation from the P factor is random. When is this a uniform permutation? How many disjoint cycles are in its cycle decomposition (which equivalently answers how many GEPP pivot movements are needed on A)? What is the longest increasing subsequence of this permutation? We will provide some statistical answers to these questions for select random matrix ensembles and transformations. For particular butterfly permutations, we will present full distributional descriptions for these particular statistics. Moreover, we introduce a random butterfly matrix ensemble that induces the Haar measure on the full 2-Sylow subgroup of the symmetric group on a set of size 2ⁿ.