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 B=(bi)i?0 B = ( b i ) i ? 0 , for instance bi?bi with b?2, and a positive integer m, let f? be the number of integers between 0 and m  having exactly ? nonzero digits in their B B -representation. It is shown that (f?)??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 (f?)??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.
log-concavity integer bases elementary symmetric polynomials binomial coefficients stirling numbers of the first kind f-vectors unimodality profiles star poset