Lognormal Degree Distribution in the Partition Graphs

TL;DR

This paper explores the structure of partition graphs, demonstrating their connectivity, analyzing degree distributions, and proposing Gray codes for higher-dimensional partitions, with evidence suggesting a lognormal distribution of degrees.

Contribution

It introduces a method to represent partitions as binary words, analyzes their graph properties, and conjectures a lognormal degree distribution, extending Gray code concepts to higher dimensions.

Findings

Partition graphs are connected for n, enabling Gray codes.

Degree distribution appears to be long-tailed lognormal.

Higher-dimensional partitions admit a 3-Gray code.

Abstract

We demonstrate a method for listing all ordinary partitions of n as binary words of length (n-1). The resulting family imbued with the hamming distance yields subgraphs of the Hamming Graphs. The existence of a 2-Gray Code for ordinary partitions follows from the fact that the graph (with the all 0s partition omitted) is 2-connected. However, the graphs fail to be hamiltonian for ordinary partitions when n > 7, ruling out the possibility of a Gray code for all such flip graphs. We further investigate the degree distribution of the graph for n, and provide computational evidence that this is a long-tailed lognormal distribution. This conjecture connects to a closely related, and much older, question of the distribution of the number of parts of a partition and the same evidence suggests that this distribution is also lognormal for large n. These methods extend to higher dimensional partitions of n which can be then written as words of length (n-1) on d + 1 letters. The resulting graphs are connected, proving that d-dimensional partitions allow a 3-Gray code.