Notes on use of generalized entropies in counting

Abstract

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-Charvát 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 Brégman's theorem. A utility of the obtained bounds is explicitly exemplified.

14 pages, no figures. Except for the style, the version 3 matches the journal version. To appear in Graphs and Combinatorics

References in corpus (2)

• Entropy and set cardinality inequalities for partition-determined functions
• Three tutorial lectures on entropy and counting

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