论文标题
一个基本矩阵的家庭完全非负性
Totally non-negativity of a family of change-of-basis matrices
论文作者
论文摘要
令$ {\ bf a} =（a_1，a_2，\ ldots，a_n）$和$ {\ bf e} =（e_1，e_1，e_2，\ ldots，e_n）$是真实的序列。用$ m _ {{{\ bf e} \ rightarrow {\ bf a}} $ $（n+1）\ times（n+1）$（m，k）$ entrix（$ m，k）$ entrix（$ m，k） $（x-a_1）\ cdots（x-a_k）$在$（x-e_1）\ cdots（x-e_m）$的扩展中作为多项式$ 1，x-a_1，\ ldots的线性组合，（x-a_1），（x-a_1）\ cdots（x-a_m）$。 By appropriate choice of ${\bf a}$ and ${\bf e}$ the matrix $M_{{\bf e}\rightarrow {\bf a}}$ can encode many familiar doubly-indexed combinatorial sequences, such as binomial coefficients, Stirling numbers of both kinds, Lah numbers and central factorial numbers. 在所有这四个示例中，$ m _ {{\ bf e} \ rightarrow {\ bf a}} $享受总非负性的属性 - 其所有正方形子膜的决定因素是非负的。这导致了一个自然的问题：一般而言，何时是$ m _ {{\ bf e} \ rightarrow {\ bf a}}} $完全非阴性？ Galvin和Pacurar在$ {\ bf e} $上发现了一个简单的条件，该条件表征了$ m _ {{\ bf e} \ rightArrow {\ bf a}} $当$ {\ bf a} $的总非阴性。在这里，我们充分扩展了此结果。对于任意的真实序列$ {\ bf a} $和$ {\ bf e} $，我们提供的条件可以在$ o（n^2）$时间中检查，该条件确定$ m _ {{\ bf e} \ rightArrow {\ rightArrow {\ rightArrow {\ bf a}} $完全不可用。当$ m _ {{{\ bf e} \ rightarrow {\ bf a}} $完全非阴性时，我们以平面网络目睹了这一点，其权重是非负的，其路径矩阵为$ m _ {{\ bf e} \ rightarrow {\ rightarrow {\ rightarrow {\ bf a} $。如果不是这样，我们就会以明确的负小调目睹这一点。
Let ${\bf a}=(a_1, a_2, \ldots, a_n)$ and ${\bf e}=(e_1, e_2, \ldots, e_n)$ be real sequences. Denote by $M_{{\bf e}\rightarrow {\bf a}}$ the $(n+1)\times(n+1)$ matrix whose $(m,k)$ entry ($m, k \in \{0,\ldots, n\}$) is the coefficient of the polynomial $(x-a_1)\cdots(x-a_k)$ in the expansion of $(x-e_1)\cdots(x-e_m)$ as a linear combination of the polynomials $1, x-a_1, \ldots, (x-a_1)\cdots(x-a_m)$. By appropriate choice of ${\bf a}$ and ${\bf e}$ the matrix $M_{{\bf e}\rightarrow {\bf a}}$ can encode many familiar doubly-indexed combinatorial sequences, such as binomial coefficients, Stirling numbers of both kinds, Lah numbers and central factorial numbers. In all four of these examples, $M_{{\bf e}\rightarrow {\bf a}}$ enjoys the property of total non-negativity -- the determinants of all its square submatrices are non-negative. This leads to a natural question: when, in general, is $M_{{\bf e}\rightarrow {\bf a}}$ totally non-negative? Galvin and Pacurar found a simple condition on ${\bf e}$ that characterizes total non-negativity of $M_{{\bf e}\rightarrow {\bf a}}$ when ${\bf a}$ is non-decreasing. Here we fully extend this result. For arbitrary real sequences ${\bf a}$ and ${\bf e}$, we give a condition that can be checked in $O(n^2)$ time that determines whether $M_{{\bf e}\rightarrow {\bf a}}$ is totally non-negative. When $M_{{\bf e}\rightarrow {\bf a}}$ is totally non-negative, we witness this with a planar network whose weights are non-negative and whose path matrix is $M_{{\bf e}\rightarrow {\bf a}}$. When it is not, we witness this with an explicit negative minor.
