In eukaryotic genomes, histone-DNA complexes called nucleosomes function to compact DNA into chromatin and to regulate access to it both by simple physical occlusion and by providing the substrate for numerous covalent epigenetic tags. While nucleosome positions in vitro are determined by genomic sequence alone, competition with other DNA-binding factors and the action of chromatin remodeling enzymes add an extra layer of complexity in living cells, making predictions of chromatin structure a computational challenge. To overcome this challenge, we have developed a DNA mechanics model for the sequence dependence of DNA bending energies, and validated it against a collection of in vitro free energies of nucleosome formation and a nucleosome crystal structure. Surprisingly, the strongest positioning signal in vivo comes from the competition with other factors rather than intrinsic nucleosome sequence preferences. For example, based on genomic sequence alone our model predicts that functional transcription factor binding sites tend to be covered by nucleosomes, but are uncovered in vivo because functional sites cluster within a single nucleosome footprint, making transcription factors bind cooperatively. Our approach distinguishes multiple ways in which genomic sequence influences nucleosome positions, and thus provides alternative explanations for several genome-wide experimental findings.