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 (LaAlO /SrTiO ) interface. The waveguide conductance follows a characteristic sequence within Pascal's triangle: (1, 3, 6, 10, 15, …) ⋅ , where is the electron charge and is the Planck constant. This behavior is consistent with the existence of a family of degenerate quantum liquids formed from bound states of = 2, 3, 4, … electrons. Our experimental setup could provide a setting for solid-state analogs of a wide range of composite fermionic phases.