One-dimensional electronic systems can support exotic collective phases because of the enhanced role of electron correlations. We describe the experimental observation of a series of quantized conductance steps within strongly interacting electron waveguides formed at the lanthanum aluminate-strontium titanate (LaAlO3/SrTiO3) interface. The waveguide conductance follows a characteristic sequence within Pascal's triangle: (1, 3, 6, 10, 15, ...). e(2)/h, where e is the electron charge and h is the Planck constant. This behavior is consistent with the existence of a family of degenerate quantum liquids formed from bound states of n = 2, 3, 4,. electrons. Our experimental setup could provide a setting for solid-state analogs of a wide range of composite fermionic phases.