Confirming Two Conjectures of Su and Wang

Details Confirming Two Conjectures of Su and Wang Confirming Two Conjectures of Su and Wang

2009-01-04

arxiv.org/abs/0901.0385v1

Adv. in Appl. Math. 43 (2009) 317--322

Mathematics

Combinatorics

8 pages, 1 figure, tentatively accepted by adv. in appl. math

Scientific paper

10.1016/j.aam.2008.12.004

Two conjectures of Su and Wang (2008) concerning binomial coefficients are proved. For $n\geq k\geq 0$ and $b>a>0$, we show that the finite sequence $C_j=\binom{n+ja}{k+jb}$ is a P\'{o}lya frequency sequence. For $n\geq k\geq 0$ and $a>b>0$, we show that there exists an integer $m\geq 0$ such that the infinite sequence $\binom{n+ja}{k+jb}, j=0, 1,...$, is log-concave for $0\leq j\leq m$ and log-convex for $j\geq m$. The proof of the first result exploits the connection between total positivity and planar networks, while that of the second uses a variation-diminishing property of the Laplace transform.

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