积和式及其计算

摘要

为了更好地用矩阵来描述组合问题，我们引入1个矩阵置换相抵下的不变量——积和式。积和式的概念在1812年由Binet和Cauchy提出的。积和式是矩阵的1个重要参数，有深刻的组合意义，在组合理论中经常将积和式与其他参数建立联系，它类似于矩阵的行列式，但又有很大的区别。
本文给出了积和式的定义如下：设 是 × 矩阵（   ），则称和式 为 的积和式（permanent）,这里 表示｛ ｝中所有 元排列的集合。
本文中详细阐述了积和式、 矩阵积和式的1些性质。在积和式的计算方面，阐述了利用Ryser定理计算积和式 的传统方法；利用正行列式得到两类 矩阵积和式，并给出其两种类型的组合应用，其后，利用正行列式建立了计算积和式 的另1种计算理论；最后还给出了关于双随机矩阵的两个问题的计算证明。

关键词:积和式；Ryser定理； 矩阵；双随机矩阵；应用

Abstract

In order to describe the question of combination with matrix better,  We introduce a constant in replacement and balance out of matrix—— Permanent. The concept of permanent set up by Binet and Cauchy in 1812. It is an important parameter of matrix with profound significance of combination. It often connects permanent with other parameters in theory of combination. It is similar to the determinant of matrix, but there are very great differences.
Define the permanent as follows: It is supposed that   is   × matrix（   ）,so claim the permanent  as the permanent of  , Here   is all  —Permutation of｛ ｝.
The text described some properties of permanent、  matrix permanent 。At calculation for permanent, it described the tradition method of utilization Ryser theorem to calculate permanent  ，Utilize the positive determinant to receive two kinds of   matrix permanent, Provide its two types association application; Thereafter, it set up another kind of calculation theory of Calculation permanent that still utilize the positive determinant; finally, provide the identifications of two questions about bistochastic matrix.

说明：中有些数学符号是编辑器编辑而成，网页上无法显示或者显示格式错误，给您带来不便请谅解。