The Burrows-Wheeler transform [1] is one of the mainstays of lossless data compression. In most cases, its output is fed to Move to Front or other variations of symbol ranking compression. One of the main open problems [2] is to establish whether Move to Front, or more in general symbol ranking compression, is an essential part of the compression process. We settle this question positively by providing a new class of Burrows-Wheeler algorithms that use optimal partitions of strings, rather than symbol ranking, for the additional step. Our technique is a quite surprising specialization to strings of partitioning techniques devised by Buchsbaum et al. [3] for two-dimensional table compression. Following Manzini [4], we analyze two algorithms in the new class, in terms of the k-th order empirical entropy of a string and, for both algorithms, we obtain better compression guarantees than the ones reported in [4] for Burrows-Wheeler algorithms that use Move to Front.