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Abstract We give a purely combinatorial proof of the positivity of the stabilized forms of the generalized exponents associated to each classical root system. In finite type $$A_{n-1}$$, we rederive the description of the generalized exponents in terms of crystal graphs without using the combinatorics of semistandard tableaux or the charge statistic. In finite type $$C_{n}$$, we obtain a combinatorial description of the generalized exponents based on the so-called distinguished vertices in crystals of type $$A_{2n-1}$$, which we also connect to symplectic King tableaux. This gives a combinatorial proof of the positivity of Lusztig $$t$$-analogs associated to zero-weight spaces in the irreducible representations of symplectic Lie algebras. We also present three applications of our combinatorial formula and discuss some implications to relating two type $$C$$ branching rules. Our methods are expected to extend to the orthogonal types.