The exploration of molecules with inverted E(S 1 -T 1 ) is essential for achieving a new generation of organic light-emitting diodes (OLEDs). The molecules must have suitable excitedstate dynamics and oscillator strengths to optimize the fluorescence rates. Combining these photophysical properties will be the key challenge in the years to come. Here, we focus on establishing rational design rules for molecules with inverted singlet-triplet gaps by studying the vertical excitation energies. To expand the library of motifs of the non-alternant hydrocarbon class, we screen the complete set of up to four fused 5-, 6-, and 7-membered rings. This constitutes 187 molecular cores which have been enumerated by Balaban and Randić. We uncover a substituent strategy that enables the stabilization of high-symmetry cores in pentalene, s-indacene and indeno[1,2,3-ef ]heptalene, and several other non-alternant cores. We identify many new molecules with inverted singlet-triplet gaps (E(S 1 -T 1 ) < 0) and higher-lying excited-state Hund's rule violations (E(S 1 -T 2 ) < 0), among which several synthesized examples exist.