The Twelvefold Way of Non-Sequential Lossless Compression

TL;DR

This paper explores lossless compression limits for various combinatorial invariances classified by the twelvefold way, providing explicit calculations and comparisons for different distributions.

Contribution

It introduces a comprehensive framework for lossless compression across twelve combinatorial invariance settings, extending beyond sequential data.

Findings

Explicit compression limits computed for all twelve settings.

Quantitative comparisons reveal differences among invariance types.

Provides a unified approach to non-sequential lossless compression.

Abstract

Many information sources are not just sequences of distinguishable symbols but rather have invariances governed by alternative counting paradigms such as permutations, combinations, and partitions. We consider an entire classification of these invariances called the twelvefold way in enumerative combinatorics and develop a method to characterize lossless compression limits. Explicit computations for all twelve settings are carried out for i.i.d. uniform and Bernoulli distributions. Comparisons among settings provide quantitative insight.