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

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...

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Bibliographic Details
Published in:Journal of statistical planning and inference 2002-02, Vol.101 (1), p.33-48
Main Author: Charalambides, Ch.A.
Format: Article
Language:English
Subjects:
Binomial hit polynomial
Exact sciences and technology
Finite differences
Mathematics
Newcomb boards
Permutations by falls
Sciences and techniques of general use
Trapezoidal boards
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Summary: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.
ISSN:0378-3758
1873-1171
DOI:10.1016/S0378-3758(01)00151-3