A Gray path on binary partitions

TL;DR

This paper constructs a Gray sequence for binary partitions of integers, enabling efficient, constant-time computation of successive partitions and directly mapping sequence indices to partitions, addressing a question by Donald Knuth.

Contribution

It introduces a novel Gray sequence on binary partitions with a simple local rule for successor computation and provides a direct bijection between sequence position and partitions.

Findings

Gray sequence on binary partitions constructed

Successor can be found in constant time

Direct bijection between sequence index and partition established

Abstract

A binary partition of a positive integer $n$ is a partition of $n$ in which each part has size a power of two. In this note we first construct a Gray sequence on the set of binary partitions of $n$. This is an ordering of the set of binary partitions of each $n$ (or of all $n$) such that adjacent partitions differ by one of a small set of elementary transformations; here the allowed transformatios are replacing $2^k+2^k$ by $2^{k+1}$ or vice versa (or addition of a new +1). Next we give a purely local condition for finding the successor of any partition in this sequence; the rule is so simple that successive transitions can be performed in constant time. Finally we show how to compute directly the bijection between $k$ and the $k$th term in the sequence. This answers a question posed by Donald Knuth in section 7.2.1 of The Art of Computer Programming.