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Log-concavity and LC-positivity | | Yi Wang
; Yeong-Nan Yeh
; | | Date: |
8 Apr 2005 | | Subject: | Combinatorics MSC-class: 05A20; 15A04 | math.CO | | Abstract: | A triangle ${a(n,k)}_{0le kle n}$ of nonnegative numbers is LC-positive if for each $r$, the sequence of polynomials $sum_{k=r}^n a(n,k)q^k$ 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 $z_n=sum_{k=0}^{n} a(n,k)x_k$; 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 $z_{n}=sum_{k=0}^{n} a(n,k)x_ky_{n-k}$. 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 Pemantle on characteristics of negative dependence. | | Source: | arXiv, math.CO/0504164 | | Services: | Forum | Review | PDF | Favorites | |
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