发表机构
The George Washington University（乔治华盛顿大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对带状Toeplitz矩阵，提出两种构造性Laplace展开方法，得到行列式与积和式的极小阶递推，并精确分类状态图，推广至循环情形。
AI 中文摘要
对于固定的非负整数$m_1,m_2$，设$A_n=(a_{j-i})_{i,j=1}^n$为具有下、上半带宽$m_1$和$m_2$的Toeplitz矩阵的前$n\times n$主子阵。我们给出了两种构造性的Laplace展开方法，用于推导$\det A_n$和$\perm A_n$的标量递推。递增行方法消去固定族的边界余子式，并对两个序列给出至多$d=\binom{m_1+m_2}{m_1}$阶的递推。行列方法递归地闭合归一化边界余子式，并将其封装在稀疏转移矩阵中。其可达状态被精确分类：第$j$层由$[m_1]$和$[m_2]$的一对$j$子集索引。因此转移维数为$d$，我们得到非零转移数的显式公式。对于行列式，递增行构造的互补余子式是经典复合伴随表示的坐标。独立构造的行列转移具有Widom特征多项式，并且通常与复合转移相似。因此，对于无限制的固定带宽行列式族，$d$阶递推通常是极小阶的。对于积和式，相同的状态图给出二项式上界，但没有一般的极小性断言。五对角情形恢复了Sweet的六阶行列式递推及其积和式类比，而单上对角族给出闭合的标量递推和有理生成函数。位置相关的带宽权重保持有限状态图，但将常数转移替换为余循环。对于循环行列式，Fourier对角化产生符号根的所有子集乘积，以及度为$2^{m_1+m_2}$的通常极小的零化多项式，对应于从单一外积度到完整外代数的过渡。
英文摘要
For fixed nonnegative integers $m_1,m_2$, let $A_n=(a_{j-i})_{i,j=1}^n$ be the leading $n\times n$ section of a Toeplitz matrix with lower and upper semibandwidths $m_1$ and $m_2$. We give two constructive Laplace-expansion methods for scalar recurrences of $\det A_n$ and $\perm A_n$. The increasing-rows method eliminates a fixed family of boundary cofactors and gives recurrence order at most $d=\binom{m_1+m_2}{m_1}$ for both sequences. The row-column method closes normalized boundary minors recursively and packages them in a sparse transfer matrix. Its reachable states are classified exactly: level $j$ is indexed by a pair of $j$-subsets of $[m_1]$ and $[m_2]$. Hence the transfer dimension is $d$, and we obtain an explicit formula for the number of nonzero transitions. For determinants, the complementary cofactors of the increasing-rows construction are coordinates of the classical compound companion representation. The independently constructed row-column transfer has the Widom characteristic polynomial and is generically similar to the compound transfer. Thus the order $d$ recurrence is generically minimal for the unrestricted fixed-band determinant family. For permanents the same state graph gives the binomial upper bound, without a general minimality claim. The pentadiagonal case recovers Sweet's order-six determinant recurrence and its permanent analogue, while the one-superdiagonal family gives closed scalar recurrences and rational generating functions. Position-dependent band weights preserve the finite state graph but replace the constant transfer by a cocycle. For cyclic determinants, Fourier diagonalization produces all subset products of the symbol roots and a generically minimal annihilator of degree $2^{m_1+m_2}$, corresponding to the passage from one exterior degree to the full exterior algebra.
