On the log-concavity of sequences arising from integer bases Original Research Article

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 image, for instance bi≔bi with b⩾2, and a positive integer m, let fℓ be the number of integers between 0 and m having exactly ℓ nonzero digits in their image-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.