Enumerative combinatorics
Sperner theory
A property of colored complexes and their duals
Discrete Mathematics - Special issue on Selected Topics in Discrete Mathematics conferences
Concrete Math
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The main result of the paper establishes the strong log-concavity of certain sequences arising from representation of positive integers with respect to some integer basis. More precisely, given an integer basis B=(bi)i≥0, for instance bi:=bi with b ≥ 2, and a positive integer m, let ft be the number of integers between 0 and m having exactly l nonzero digits in their B-representation. It is shown that (fl)l≥0 is log-concave and some estimates for the peaks of these sequences are given. This theorem is indeed an inequality for elementary symmetric polynomials. It can be specialized to give the log-concavity of sequences of sums of special numbers, such as binomial coefficients, Stirling numbers of the first kind or their q-analogs. These sequences (fl)l≥0 can also be seen as f-vectors of compressed subsets in direct (poset) product of stars, where the compression is relative to the reverse-lexicographic order.