On the log-concavity of sequences arising from integer bases

✍ Scribed by Philippe Pitteloud

• Book ID: 104294222
• Publisher: Elsevier Science
• Year: 2004
• Tongue: English
• Weight: 337 KB
• Volume: 138
• Category: Article
• ISSN: 0166-218X
• DOI: 10.1016/s0166-218x(03)00292-0

✦ Synopsis

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 := b i 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-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 coe cients, Stirling numbers of the ÿrst 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.