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The challenge of understanding strongly interacting composite fermionic phases of matter spans many fields in physics, ranging from neutron stars to solid-state materials to quark-gluon plasmas. We report experimental evidence of a new family of degenerate quantum liquids formed from bound states of n = 2, 3, 4, … electrons, which are stabilized within quasi-one-dimensional electron waveguides formed at the LaAlO₃/SrTiO₃ interface. The key signature of this phase is the existence of quantized conduction that follows a characteristic sequence within Pascal’s triangle: (1, 3, 6, 10, 15,…), where is the electron charge and is the Planck constant. The ability to create and investigate composite fermionic phases opens new avenues for the investigation of strongly correlated quantum matter.