Log-concavity and LC-positivity

JOURNAL OF COMBINATORIAL THEORY SERIES A

卷:114

Article

Wang, Yi ; Yeh, Yeong-Nan

关键词: sequences;    linear transformations;    convolutions;    log-concavity;    q-log-concavity;    LC-positivity;   

DOI  :  10.1016/j.jcta.2006.02.001

来源: Elsevier

【 摘 要 】

A triangle {a(n, k)}(0 <= k <= n) of nonnegative numbers is LC-positive if for each r, the sequence of polynomials Sigma(n)(k=r)a(n,k)q(k) is q-log-concave. It is double LC-positive if both triangles {a(n,k)} and {a(n, n-k)} are LC-positive. We show that if {a(n, k)} is LC-positive then the log-concavity of the sequence {x(k)} implies that of the sequence {z(n)} defined by z(n) = Sigma(n)(k=0) a(n, k)x(k), and if {a(n, k)} is double LC-positive then the log-concavity of sequences {x(k)} and {y(k)} implies that of the sequence {z(n)} defined by z(n) = Sigma(n)(k=0) a(n, k)x(k gamma n-k). Examples of double LC-positive triangles include the constant triangle and the Pascal triangle. We also give a generalization of a result of Liggett that is used to prove a conjecture of Permantle on characteristics of negative dependence. (c) 2006 Elsevier Inc. All rights reserved.

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