q-ballot数的q-对数凹性  

The q-log-concavity of q-ballot Numbers

作　　者：刘鑫淼 侯江霞[1] 刘凤霞[1] LIU Xinmiao;HOU Jiangxia;LIU Fengxia(College of Mathematics and Systems Science,Xinjiang University,Urumqi,Xinjiang,830046,P.R.China)

机构地区：[1]新疆大学数学与系统科学学院,新疆乌鲁木齐830046

出　　处：《数学进展》2024年第2期243-249,共7页Advances in Mathematics（China）

基　　金：国家自然科学基金(No.11961067)。

摘　　要：Carlitz和Riordan引入了ballot数的q模拟f_(q)(n,k).本文利用f_(q)(n,k)的组合解释,通过构造单射的方法证明f_(q)(n,k)关于n和关于k都具有q-对数凹性,即关于q的多项式f_(q)(n,k)^(2)-f_(q)(n+1,k)f_(q)(n-1,k)和f_(q)(n,k)2-f_(q)(n,k+1)f_(q)(n,k-1)对于0<k<n都具有非负系数.Carlitz and Riordan introduced a q-analogue of ballot numbers f_(q)(n,k).In this paper,by using the combinatorial interpretation of f_(q)(n,k)and constructing injections,we prove that f_(q)(n,k)is q-log-concave with respect to n and k,that is,all the coefficients of the polynomials f_(q)(n,k)2−f_(q)(n+1,k)f_(q)(n−1,k)and f_(q)(n,k)2−f_(q)(n,k+1)f_(q)(n,k−1)are nonnegative for 0<k<n.

关 键 词：q-对数凹性 q-ballot数 格路 逆序数 

分 类 号：O157.1[理学—数学]