##### Department of Mathematics,

University of California San Diego

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### Math 269 - Combinatorics

## Glenn Tesler

#### UCSD

## Multi de Bruijn Sequences

##### 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 $\{0,1\}$, 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 $m=1$.

Organizer: Jeff Remmel

### April 11, 2017

### 4:00 PM

### AP&M 7321

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