# Some Enumeration Problems in the Duplication-Loss Model of Genome Rearrangement

## TL;DR

This paper explores the mathematical properties of tandem-duplication-random-loss (TDRL) genome rearrangements, providing insights into their structure and potential applications in DNA data storage and error correction.

## Contribution

It determines the sizes of TDRL 'balls' and their intersections in the permutation space, advancing the understanding of TDRL operations and their mirror variants.

## Key findings

- Cardinality of TDRL balls of radius one is established.
- Maximum intersection size of two TDRL balls is calculated.
- Results have implications for DNA data storage and error correction.

## Abstract

Tandem-duplication-random-loss (TDRL) is an important genome rearrangement operation studied in evolutionary biology. This paper investigates some of the formal properties of TDRL operations on the symmetric group (the space of permutations over an $ n $-set). In particular, the cardinality of `balls' of radius one in the TDRL metric, as well as the cardinality of the maximum intersection of two such balls, are determined. The corresponding problems for the so-called mirror (or palindromic) TDRL rearrangement operations are also solved. The results represent an initial step in the study of error correction and reconstruction problems in this context and are of potential interest in DNA-based data storage applications.