On the log-concavity of sequences arising from integer bases

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 <f>B=(bi)i⩾0</f>, for instance <f>bi≔bi</f> with <f>b⩾2</f>, and a positive integer <f>m</f>, let <f>fℓ</f> be the number of integers between 0 and <f>m</f> having exactly <f>ℓ</f> nonzero digits in their <f>B</f>-representation. It is shown that <f>(fℓ)ℓ⩾0</f> 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 <f>q</f>-analogs. These sequences <f>(fℓ)ℓ⩾0</f> can also be seen as <f>f</f>-vectors of compressed subsets in direct (poset) product of stars, where the compression is relative to the reverse-lexicographic order. [Copyright &y& Elsevier]