## Margaret Archibald ; Aubrey Blecher ; Charlotte Brennan ; Arnold Knopfmacher ; Stephan Wagner et al. - The number of distinct adjacent pairs in geometrically distributed words

dmtcs:5686 - Discrete Mathematics & Theoretical Computer Science, January 28, 2021, vol. 22 no. 4 - https://doi.org/10.23638/DMTCS-22-4-10
The number of distinct adjacent pairs in geometrically distributed words

Authors: Margaret Archibald ; Aubrey Blecher ; Charlotte Brennan ; Arnold Knopfmacher ; Stephan Wagner ; Mark Ward

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 $n \to \infty$.

Volume: vol. 22 no. 4
Section: Analysis of Algorithms
Published on: January 28, 2021
Accepted on: November 10, 2020
Submitted on: August 12, 2019
Keywords: Mathematics - Combinatorics,05A15, 05A05