Notes on use of generalized entropies in counting

We address an idea of applying generalized entropies in counting problems. First, we consider some entropic properties that are essential for such purposes. Using the $\alpha$-entropies of Tsallis-Havrda-Charv\'{a}t type, we derive several results connected with Shearer's lemma. In particular, we derive upper bounds on the maximum possible cardinality of a family of $k$-subsets, when no pairwise intersections of these subsets may coincide. Further, we revisit the Minc conjecture. Our approach leads to a family of one-parameter extensions of Br\'{e}gman's theorem. A utility of the obtained bounds is explicitly exemplified.