Geodesic
The number of distinct adjacent pairs in geometrically distributed words
Discrete mathematics & theoretical computer science, Tome 22 (2020-2021) no. 4

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A sequence of geometric random variables of length $n$ is a sequence of $n$ independent and identically distributed geometric random variables ($\Gamma_1, \Gamma_2, \dots, \Gamma_n$) where $\mathbb{P}(\Gamma_j=i)=pq^{i-1}$ for $1~\leq~j~\leq~n$ with $p+q=1.$ We study the number of distinct adjacent two letter patterns in such sequences. Initially we directly count the number of distinct pairs in words of short length. Because of the rapid growth of the number of word patterns we change our approach to this problem by obtaining an expression for the expected number of distinct pairs in words of length $n$. We also obtain the asymptotics for the expected number as $n \to \infty$. 
@article{DMTCS_2020_22_4_a12,
     author = {Archibald, Margaret and Blecher, Aubrey and Brennan, Charlotte and Knopfmacher, Arnold and Wagner, Stephan and Ward, Mark},
     title = {The number of distinct adjacent pairs in geometrically distributed words},
     journal = {Discrete mathematics & theoretical computer science},
     publisher = {mathdoc},
     volume = {22},
     number = {4},
     year = {2020-2021},
     doi = {10.23638/DMTCS-22-4-10},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.23638/DMTCS-22-4-10/}
}
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Archibald, Margaret; Blecher, Aubrey; Brennan, Charlotte; Knopfmacher, Arnold; Wagner, Stephan; Ward, Mark. The number of distinct adjacent pairs in geometrically distributed words. Discrete mathematics & theoretical computer science, Tome 22 (2020-2021) no. 4. doi: 10.23638/DMTCS-22-4-10

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