  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.
