Lamellae-forming polystyrene-block-poly(methyl methacrylate) (PS-b-PMMA) films, with bulk period L-0, were directed to assemble on lithographically nanopatterned surfaces. The chemical pattern was comprised of "guiding" stripes of cross-linked polystyrene (X-PS) or poly(methyl methacrylate) (X-PMMA) mats, with width W, and interspatial "background" regions of a random copolymer brush of styrene and methyl methacrylate (P(S-r-MMA)). The fraction of styrene (f) in the brush was varied to control the chemistry of the background regions. The period of the pattern was L-s. After assembly, the density of the features (domains) in the block copolymer film was an integer multiple (n) of the density of features of the chemical pattern, where n = L-s/L-0. The quality of the assembled PS-b-PMMA films into patterns of dense lines as a function of n, W/L-0, and f was analyzed with top-down scanning electron microscopy. The most effective background chemistry for directed assembly with density multiplication corresponded to a brush chemistry (f) that minimized the interfacial energy between the background regions and the composition of the film overlying the background regions. The three-dimensional structure of the domains within the film was investigated using cross-sectional SEM and Monte Carlo simulations of a coarse-grained model and was found most closely to resemble perpendicularly oriented lamellae when W/L-0 similar to 0.5-0.6. Directed self-assembly with density multiplication (n = 4) and W/L-0 = 1 or 1.5 yields pattern of high quality, parallel linear structures on the top surface of the assembled films, but complex, three-dimensional structures within the film.