Log‐Concave and Unimodal Sequences in Algebra, Combinatorics, and Geometry a
Log‐Concave and Unimodal Sequences in Algebra, Combinatorics, and Geometry a
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DOI：
10.1111/j.1749-6632.1989.tb16434.x
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发表时间：
1989-12
影响因子：
5.2
通讯作者：
R. Stanley
中科院分区：
文献类型：
--
作者：
R. Stanley
序列 a,, a,, 。 。 。如果对于某些 0 s j _c n 我们有 a, 5 a , 5 ，则实数的 , a 被称为单峰。 。 5 ai 2 a,,, 2 。 。 2 a,, 如果 a: 2 a, ,a ,+ , 对于所有 1 5 i 5 n 1，则被称为对数凹（或简称对数凹）。显然，正项的对数凹序列是单峰的。假设序列 a,, a,, 。 。 。如果不存在满足 a, f 0, a, = 0, ak # 0 的整数 0 5 i < j < k 5 n，则 a, 没有内部零点。那么实际上没有内部零点的非负对数凹序列是单峰的。序列 a,, a , , . 。 。 , a, 称为对称，如果 a, = a,-, 对于 0 5 i 5 n。因此，对称单峰序列 a,, a, , 。 。 。 , a,, 在中间项（n 偶数）或中间两项（n 奇数）处具有最大值。我们也称多项式a,+a,4+。 。 + an$ 如果其序列 a,, a,, 则具有特定属性（例如单峰、对数凹或对称）。 。 。系数的 , a, 具有该属性。我们这里的目的是调查显示序列是对数凹或单峰的极其丰富的方法。对于每种方法，我们将举例说明其对组合定义序列的适用性，这些序列自然产生于代数、组合学和几何问题。然而，我们并不试图全面介绍该领域所做的所有工作。
A sequence a,, a,, . . . , a, of real numbers is said to be unimodal if for some 0 s j _c n we have a, 5 a , 5 . . 5 ai 2 a,,, 2 . . 2 a,, and is said to be logarithmically concave (or log-concave for short) if a: 2 a, ,a ,+ , for all 1 5 i 5 n 1. Clearly a log-concave sequence of positive terms is unimodal. Let us say that the sequence a,, a,, . . . , a, has no internal zeros if there do not exist integers 0 5 i < j < k 5 n satisfying a, f 0, a, = 0, ak # 0. Then in fact a nonnegative log-concave sequence with no internal zeros is unimodal. The sequence a,, a , , . . . , a, is called symmetric if a, = a,-, for 0 5 i 5 n. Thus a symmetric unimodal sequence a,, a, , . . . , a,, has its maximum at the middle term ( n even) or middle two terms (n odd). We also say that a polynomial a, + a,4 +. . + an$ has a certain property (such as unimodal, log-concave, or symmetric) if its sequence a,, a,, . . . , a, of coefficients has that property. Our object here is to survey the surprisingly rich variety of methods for showing that a sequence is log-concave or unimodal. For each method we will give examples of its applicability to combinatorially defined sequences that arise naturally from problems in algebra, combinatorics, and geometry. We make no attempt, however, to give a comprehensive account of all work done in this area.