Limit Theorems for the Length of the Longest Common Subsequence of Mallows Permutations
The Mallows measure is measure on permutations which was introduced by Mallows in connection with ranking problems in statistics. Under this measure, the probability of a permutation $\pi$ is proportional to $q^{Inv(\pi)}$ where $q$ is a positive parameter and $Inv(\pi)$ is the number of inversions...
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| creator | Banerjee, Naya Jin, Ke |
| description | The Mallows measure is measure on permutations which was introduced by
Mallows in connection with ranking problems in statistics. Under this measure,
the probability of a permutation $\pi$ is proportional to $q^{Inv(\pi)}$ where
$q$ is a positive parameter and $Inv(\pi)$ is the number of inversions in
$\pi$. We consider the length of the longest common subsequence (LCS) of two
independently permutations drawn according to $\mu_{n,q}$ and $\mu_{n,q'}$ for
some $q,q' >0$.
We show that when $0 |
| doi_str_mv | 10.48550/arxiv.1908.05246 |
| format | Article |
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Mallows in connection with ranking problems in statistics. Under this measure,
the probability of a permutation $\pi$ is proportional to $q^{Inv(\pi)}$ where
$q$ is a positive parameter and $Inv(\pi)$ is the number of inversions in
$\pi$. We consider the length of the longest common subsequence (LCS) of two
independently permutations drawn according to $\mu_{n,q}$ and $\mu_{n,q'}$ for
some $q,q' >0$.
We show that when $0<q,q'<1$, the limiting law of the LCS is Gaussian. In the
regime that $n(1-q) \to \infty$ and $n(1-q') \to \infty$ we show a weak law of
large numbers for the LCS. These results extend the results of \cite{Basu} and
\cite{Naya} showing weak laws and a limiting law for the distribution of the
longest increasing subsequence to showing corresponding results for the longest
common subsequence.</description><identifier>DOI: 10.48550/arxiv.1908.05246</identifier><language>eng</language><subject>Mathematics - Combinatorics ; Mathematics - Probability</subject><creationdate>2019-08</creationdate><rights>http://arxiv.org/licenses/nonexclusive-distrib/1.0</rights><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><link.rule.ids>228,230,776,881</link.rule.ids><linktorsrc>$$Uhttps://arxiv.org/abs/1908.05246$$EView_record_in_Cornell_University$$FView_record_in_$$GCornell_University$$Hfree_for_read</linktorsrc><backlink>$$Uhttps://doi.org/10.48550/arXiv.1908.05246$$DView paper in arXiv$$Hfree_for_read</backlink></links><search><creatorcontrib>Banerjee, Naya</creatorcontrib><creatorcontrib>Jin, Ke</creatorcontrib><title>Limit Theorems for the Length of the Longest Common Subsequence of Mallows Permutations</title><description>The Mallows measure is measure on permutations which was introduced by
Mallows in connection with ranking problems in statistics. Under this measure,
the probability of a permutation $\pi$ is proportional to $q^{Inv(\pi)}$ where
$q$ is a positive parameter and $Inv(\pi)$ is the number of inversions in
$\pi$. We consider the length of the longest common subsequence (LCS) of two
independently permutations drawn according to $\mu_{n,q}$ and $\mu_{n,q'}$ for
some $q,q' >0$.
We show that when $0<q,q'<1$, the limiting law of the LCS is Gaussian. In the
regime that $n(1-q) \to \infty$ and $n(1-q') \to \infty$ we show a weak law of
large numbers for the LCS. These results extend the results of \cite{Basu} and
\cite{Naya} showing weak laws and a limiting law for the distribution of the
longest increasing subsequence to showing corresponding results for the longest
common subsequence.</description><subject>Mathematics - Combinatorics</subject><subject>Mathematics - Probability</subject><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2019</creationdate><recordtype>article</recordtype><sourceid>GOX</sourceid><recordid>eNotz8tKxDAYBeBsXMjoA7gyL9CaJmnSLKV4g4qCBZclTf5MC02jSerl7WVmXB0OHA58CF1VpORNXZMbHX_mr7JSpClJTbk4R-_d7OeM-wlCBJ-wCxHnCXAH6z5POLhTC-seUsZt8D6s-G0bE3xusBo4LJ71soTvhF8h-i3rPIc1XaAzp5cEl_-5Q_39Xd8-Ft3Lw1N72xVaSFEYXjVWUlZbw1lDKZOG89poSwVhxHKppHQVkNE5pUAZEEIJJ0ddKQugge3Q9en2KBs-4ux1_B0OwuEoZH-Uskx-</recordid><startdate>20190814</startdate><enddate>20190814</enddate><creator>Banerjee, Naya</creator><creator>Jin, Ke</creator><scope>AKZ</scope><scope>GOX</scope></search><sort><creationdate>20190814</creationdate><title>Limit Theorems for the Length of the Longest Common Subsequence of Mallows Permutations</title><author>Banerjee, Naya ; Jin, Ke</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-a676-c418d7235dc4382237c445cad26030d47977f1e0bff99e9ce6696f7ba19deeae3</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2019</creationdate><topic>Mathematics - Combinatorics</topic><topic>Mathematics - Probability</topic><toplevel>online_resources</toplevel><creatorcontrib>Banerjee, Naya</creatorcontrib><creatorcontrib>Jin, Ke</creatorcontrib><collection>arXiv Mathematics</collection><collection>arXiv.org</collection></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext_linktorsrc</fulltext></delivery><addata><au>Banerjee, Naya</au><au>Jin, Ke</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Limit Theorems for the Length of the Longest Common Subsequence of Mallows Permutations</atitle><date>2019-08-14</date><risdate>2019</risdate><abstract>The Mallows measure is measure on permutations which was introduced by
Mallows in connection with ranking problems in statistics. Under this measure,
the probability of a permutation $\pi$ is proportional to $q^{Inv(\pi)}$ where
$q$ is a positive parameter and $Inv(\pi)$ is the number of inversions in
$\pi$. We consider the length of the longest common subsequence (LCS) of two
independently permutations drawn according to $\mu_{n,q}$ and $\mu_{n,q'}$ for
some $q,q' >0$.
We show that when $0<q,q'<1$, the limiting law of the LCS is Gaussian. In the
regime that $n(1-q) \to \infty$ and $n(1-q') \to \infty$ we show a weak law of
large numbers for the LCS. These results extend the results of \cite{Basu} and
\cite{Naya} showing weak laws and a limiting law for the distribution of the
longest increasing subsequence to showing corresponding results for the longest
common subsequence.</abstract><doi>10.48550/arxiv.1908.05246</doi><oa>free_for_read</oa></addata></record> |
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| subjects | Mathematics - Combinatorics Mathematics - Probability |
| title | Limit Theorems for the Length of the Longest Common Subsequence of Mallows Permutations |
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