Frogs, hats and common subsequences

Fuente: arXiv
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Auteurs principaux: Briggs, Joseph, Parker, Alex, Schwieder, Coy, Wells, Chris
Format: Preprint
Publié: 2024
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author Briggs, Joseph
Parker, Alex
Schwieder, Coy
Wells, Chris
author_facet Briggs, Joseph
Parker, Alex
Schwieder, Coy
Wells, Chris
contents Write $W^{(n)}$ to mean the $n$-letter word obtained by repeating a fixed word $W$ and let $R_n$ denote a uniformly random $n$-letter word sampled from the same alphabet as $W$. We are interested in the average length of the longest common subsequence between $W^{(n)}$ and $R_n$, which is known to be $γ(W)\cdot n+o(n)$ for some constant $γ(W)$. Bukh and Cox recently developed an interacting particle system, dubbed the frog dynamics, which can be used to compute the constant $γ(W)$ for any fixed word $W$. They successfully analyzed the simplest case of the frog dynamics to find an explicit formula for the constants $γ(12\cdots k)$. We continue this study by using the frog dynamics to find an explicit formula for the constants $γ(12\cdots kk\cdots 21)$. The frog dynamics in this case is a variation of the PushTASEP on the ring where some clocks are identical. Interestingly, exclusion processes with correlated clocks of this type appear to have not been analyzed before. Our analysis leads to a seemingly new combinatorial object which could be of independent interest: frogs with hats!
format Preprint
id arxiv_https___arxiv_org_abs_2404_07285
institution arXiv
publishDate 2024
record_format arxiv
spellingShingle Frogs, hats and common subsequences
Briggs, Joseph
Parker, Alex
Schwieder, Coy
Wells, Chris
Combinatorics
Probability
05A19, 60J10
Write $W^{(n)}$ to mean the $n$-letter word obtained by repeating a fixed word $W$ and let $R_n$ denote a uniformly random $n$-letter word sampled from the same alphabet as $W$. We are interested in the average length of the longest common subsequence between $W^{(n)}$ and $R_n$, which is known to be $γ(W)\cdot n+o(n)$ for some constant $γ(W)$. Bukh and Cox recently developed an interacting particle system, dubbed the frog dynamics, which can be used to compute the constant $γ(W)$ for any fixed word $W$. They successfully analyzed the simplest case of the frog dynamics to find an explicit formula for the constants $γ(12\cdots k)$. We continue this study by using the frog dynamics to find an explicit formula for the constants $γ(12\cdots kk\cdots 21)$. The frog dynamics in this case is a variation of the PushTASEP on the ring where some clocks are identical. Interestingly, exclusion processes with correlated clocks of this type appear to have not been analyzed before. Our analysis leads to a seemingly new combinatorial object which could be of independent interest: frogs with hats!
title Frogs, hats and common subsequences
topic Combinatorics
Probability
05A19, 60J10
url https://arxiv.org/abs/2404.07285