Gaussian elimination with partial pivoting (GEPP) remains the most used dense linear solver. For a n x n 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 cycles are in its disjoint cycle decomposition (which equivalently answers how many GEPP pivot movements are needed on A)? What is the length of 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 a full 2-Sylow subgroup of the symmetric group on a set of size 2ⁿ.