Linear and Circular Permutations with Limited Number of Repetitions

Abstract

The problem of linear and circular permutations of n identical objects in m boxes, where a limit l is imposed on the number of objects in a box, is considered. In the linear case, where the boxes are arranged as a row, two methods of solution are described. The first uses the partition diagram that is modified with prescribed number of zeros, and the second applies a direct combinatorial approach. Subject to constraints set on the relation between n and l, results of the combinatorial approach are used to solve the problem where there are more than one type of objects. These solutions are then extended for the case of boxes comprising subgroups that are characterized by different values of l. In the circular case, where the boxes are arranged in the form of a circle, no direct combinatorial solution in close form is available. Consequently, in the latter case, the solution relies on the use of the modified partition diagram. This solution, which involves the application of multinomial permutations on a circle, is also extended to the case of two kinds of objects.