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# Journal of Combinatorics

## Volume 8 (2017)

### Number 3

Guest Editor: Steve Butler (Iowa State University)

### Multi de Bruijn sequences

Pages: 439 – 474

DOI: http://dx.doi.org/10.4310/JOC.2017.v8.n3.a3

#### Author

#### Abstract

We generalize the notion of a de Bruijn sequence to a “multi de Bruijn sequence”: a cyclic or linear sequence that contains every $k$-mer over an alphabet of size $q$ exactly $m$ times. For example, over the binary alphabet $\lbrace 0, 1 \rbrace$, the cyclic sequence $(00010111)$ and the linear sequence $000101110$ each contain two instances of each $2$-mer $00, 01, 10, 11$. We derive formulas for the number of such sequences. The formulas and derivation generalize classical de Bruijn sequences (the case $m = 1$). We also determine the number of multisets of aperiodic cyclic sequences containing every $k$-mer exactly $m$ times; for example, the pair of cyclic sequences $(00011)(011)$ contains two instances of each $2$-mer listed above. This uses an extension of the Burrows–Wheeler Transform due to Mantaci *et al.*, and generalizes a result by Higgins for the case $m = 1$.

#### Keywords

multi de Bruijn sequences, de Bruijn graphs, Extended Burrows–Wheeler Transform, Lyndon words, Eulerian graphs

#### 2010 Mathematics Subject Classification

Primary 05C30, 68R15. Secondary 05C38, 05C45, 05C81, 68P30.

This work was supported in part by National Science Foundation grant CCF-1115206.

Received 16 August 2016

Published 21 June 2017