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We provide q -generalizations of Spivey’s Bell number formula in various settings by considering statistics on different combinatorial structures. This leads to new identities involving q -Stirling numbers of both kinds and q -Lah numbers. As corollaries, we obtain identities for both binomial and q -binomial coefficients. Our results at the same time also generalize recent r -Stirling number formulas of Mezo. Finally, we provide a combinatorial proof and refinement of Xu’s extension of Spivey’s formula to the generalized Stirling numbers of Hsu and Shiue. To do so, we develop a combinatorial interpretation for these numbers in terms of extended Lah distributions.

Generalizations of Bell number formulas of spivey and Mező