Fast Algorithm for Generating Ascending Compositions

Abstract

In this paper we give a fast algorithm to generate all partitions of a positive integer n. Integer partitions may be encoded as either ascending or descending compositions for the purposes of systematic generation. It is known that the ascending composition generation algorithm is substantially more efficient than its descending composition counterpart. Using tree structures for storing the partitions of integers, we develop a new ascending composition generation algorithm which is substantially more efficient than the algorithms from the literature.

Binary Diagrams for Storing Ascending Compositions

“…Taking into account (17), (18) and Lemma 4.1, the proof is similar to the proof of Merca [7,Theorem 5].…”

“…To do this, we can convert efficient algorithms for traversing of the partition strict binary trees. In [7], we described and analyzed Algorithms 2 and 3 for in-order traversal of the partition strict binary trees. To convert these algorithms, we use the ascending composition array representation of the partition binary diagram and the following result.…”

“…To present the performance of Algorithm 2, we use the integer p (t) (n) introduced in [7,Section 2]: the number of the partitions λ n that have the property…”

“…Using strict binary tree structures, we produced in [7] the fastest known algorithm for the generation of the partitions of n. The data structure for storing ascending compositions of n was called the partition strict binary tree and was created according to the following rule: the root of the partition strict binary tree of n is labeled with (1, n − 1), the node (x l , y l ) is the left child of the node (x, y) if and only if …”

“…Note that the partition strict binary trees were derived in [7] from tree structures presented by Lin [5] and are different from the binary tree structures introduced by Fenner and Loizou [4].…”