摘要
國際象棋中的車可直行與橫行。如果在任意形狀的棋盤上放置數個車，使得這些車不互相攻擊，則每個車必須彼此位在不同行不同列上。車多項式是指將車放置在棋盤上的方法數之生成函數。車多項式可用來解決有限制的排列的問題。因此我們希望能藉由探討一些特殊國際象棋中的車可直行與橫行。如果在任意形狀的棋盤上放置數個車，使得這些車不互相攻擊，則每個車必須彼此位在不同行不同列上。車多項式是一種放置各種不同個數的車的方法數的生成函數。車多項式可用來解決有限制的排列的問題。因此我們希望能藉由探討一些特殊棋盤的車多項式，獲得更快速解決有限制的排列的問題。 在論文中，我們主要推導並證明了四種特殊棋盤的車多項式： 1.m×n棋盤的車多項式。 2.有禁區的車多項式。 3.路徑棋盤的車多項式。 4.迴圈棋盤的車多項式。
並列摘要
In combinatorial mathematics, a rook polynomial is a generating function of the number of ways to place non-attacking rooks on a board that looks like a checker board; that is, no two rooks can be placed in the same row or same column. The term "rook polynomial" was coined by John Riordan. Despite the name's derivation from chess, the impetus for studying rook polynomials is their connection with counting the number of permutations with restricted positions. In this thesis, we mainly obtain the rook polynomials of four special boards: 1.The rook polynomial of m×n chess board. 2.The rook polynomial with restricted area 3.The rook polynomial of path chess board 4.The rook polynomial of cycle chess board