Decoding combinations of the first n integers taken k at a time

ABSTRACT

The following problem arose in connection with some studies involving game programming: the representation of any position in the game tree was formed by a sequence of k different integers selected from the first n integers. It was desired to decode any of these representations to a unique memory address such that all such addresses formed a compact group in the memory. Mathematically, the problem was to find a transformation of the representation such that any of the n!/(n - k)! sequences transformed uniquely to one of the set of integers from 0 to [n!/(n - k)! - 1]. A procedure for this is described here.