Abstract
Abstract The number of ways of placing k non-taking rooks on a Ferrers board Bn 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 Bn, 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.
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