A Naturally Ordered Enumeration of Compositions

Abstract A composition of the positive integer M , with N -parts, is a vector of N non-negative integer components the sum of which is M . The paper presents a transformation that “enumerates” (assigns serial numbers) the set of all possible compositions most densely so that their “natural” ordering is preserved. The transformation is useful for the efficient retrieval and compact storage of probabilities, or functions on probability spaces, in finite memories.