Generating All Partitions: A Comparison Of Two Encodings
Abstract
Integer partitions may be encoded as either ascending or descending compositions for the purposes of systematic generation. Many algorithms exist to generate all descending compositions, yet none have previously been published to generate all ascending compositions. We develop three new algorithms to generate all ascending compositions and compare these with descending composition generators from the literature. We analyse the new algorithms and provide new and more precise analyses for the descending composition generators. In each case, the ascending composition generation algorithm is substantially more efficient than its descending composition counterpart. We develop a new formula for the partition function p(n) as part of our analysis of the lexicographic succession rule for ascending compositions.
Keywords
Cite
@article{arxiv.0909.2331,
title = {Generating All Partitions: A Comparison Of Two Encodings},
author = {Jerome Kelleher and Barry O'Sullivan},
journal= {arXiv preprint arXiv:0909.2331},
year = {2014}
}
Comments
40 pages, 2 figures. Derived from J. Kelleher's Ph.D thesis, Encoding Partitions as Ascending Compositions, University College Cork, 2006. V2: corrected typos in RuleDesc algorithm