Log-concave and unimodal sequences in algebra, combinatorics, and geometry: An update. (English) Zbl 0813.05007

Barcelo, Hélène (ed.) et al., Jerusalem combinatorics ’93: an international conference in combinatorics, May 9-17, 1993, Jerusalem, Israel. Providence, RI: American Mathematical Society. Contemp. Math. 178, 71-89 (1994).
A sequence \((a_ 0, a_ 1,\dots, a_ d)\) of real numbers is called log-concave if \(a_ i^ 2\geq a_{i-1} a_{i+ 1}\) for \(i= 1,2,\dots, d- 1\). It is said to be unimodal if \(a_ 0\leq a_ 1\leq \cdots\leq a_{j- 1}\leq a_ j\geq a_{j+ 1}\geq\cdots\geq a_ d\), for some \(j\). In [Ann. N. Y. Acad. Sci. 576, 500-534 (1989; Zbl 0792.05008)] R. P. Stanley gave a survey of methods, problems, conjectures, and theorems concerning log-concave and unimodal sequences. The paper under review is intended to be a supplement to Stanley’s article. It contains the new developments in the subject, in particular new techniques, new results, and also new problems that have emerged. Each of Stanley’s sections (except for two, where nothing new has been contributed) has a counterpart in the paper under review. A section on total positivity is added, a subject to which the author has significantly contributed.
For the entire collection see [Zbl 0806.00023].


05A99 Enumerative combinatorics
05-02 Research exposition (monographs, survey articles) pertaining to combinatorics
05A20 Combinatorial inequalities
05A10 Factorials, binomial coefficients, combinatorial functions
05A15 Exact enumeration problems, generating functions


Zbl 0792.05008