一排列问题的推广及其计数公式

Generalization and Counting of a Certain Permutation

对“全国大学生数学建模竞赛 ( 1 997)”的一道题进行推广 :将 aij( 1≤ i≤ n,1≤ j≤ m)排成一排或一个圆圈 ,当ai1,ai2 ,… ,aim( i=1 ,2 ,… ,n)相邻时 ,则将其看作一个整体而不考虑它们之间的顺序 ,则这种排列的个数 Ln,m与 Rn,m分别为 :Ln,m=∑ni=0( -1 ) i( m!-1 ) i Cin( mn -( m -1 ) i) !,Rn,m=∑ni=0( -1 ) i( m!-1 ) i Cin( mn -( m -1 ) i-1 ) !

A problem in \%Mathematical Contest in Modeling (1997)\% is generalised as follows: Let \%a\-\{ij\} (1≤i≤n, 1≤j≤m)\% in a line or a cycle, when \%a\-\{i1\},a\-\{i2\},\:,a\-\{im\} (i=1,2,\:,n)\% are adjacent to each other, their orders are ignored. Then the number of such permutation is, respectively,  L n,m =∑ni=0(-1) i(m!-1) iC i n(mn-(m-1)i)!; R n,m =∑ni=0(-1) i(m!-1) iC i n(mn-(m-1)i-1)!.