Is there more concise representation of simplicial sets than complete enumeration of its simplices and gluings - application to ∞-categories?

Question

Is there more concise representation of simplicial sets than complete enumeration of its simplices and gluings. As special simplicial sets (e.Kan complexes) are models for the ∞-categories, then this question is also about the more concise representation of (relatively) large ∞-categories which can go beyond the geometric representation of diagrammatic representation (e.g. https://github.com/emilyriehl/yoneda). Maybe there are some generating functions which can express such combinatorial structures, or maybe there is some other ways if the simplicial set contains some regularities?