The rook numbers of Ferrers boards and the related restricted permutation numbers

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• 作者：Charalambides ; Ch.A.
• 关键词：primary ; 05A15 ; secondary ; 05A10 ; Binomial hit polynomial ; Finite differences ; Newcomb boards ; Permutations by falls ; Trapezoidal boards
• 刊名：Journal of Statistical Planning and Inference
• 出版年：2002
• 期刊代码：36_03783758
• 类别：mt
• 出版时间：February 15, 2002
• 卷：101
• 期：1-2
• 页码：33-48
• 文件大小：171 K

摘要

The number of ways of placing k non-taking rooks on a Ferrers board B_n is obtained as a finite difference of order n−k of a polynomial of order n. Then the number of permutations of the set {1,2,…,n} with k elements in restricted positions, when the board of the restricted positions is a Ferrers board B_n, is derived as a backward finite difference of order n+1 of a modified polynomial of order n. Further, triangular recurrence relations for these numbers are provided. In particular, the cases of trapezoidal and Newcomb boards are examined. As applications, the numbers of permutations of a set, without or with repetitions, and a multiset, which have a given number of falls (or rises) are deduced.