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Volume 35, Issue 1
Singly Covered Minimal Elements of Linked Partitions and Cycles of Permutations

Wenkui Guo & Feifei Niu

Commun. Math. Res., 35 (2019), pp. 75-80.

Published online: 2019-12

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  • Abstract

Linked partitions were introduced by Dykema (Dykema K J. Multilinear function series and transforms in free probability theory. $Adv$. $Math$., 2005, 208(1): 351–407) in the study of the unsymmetrized T-transform in free probability theory. Permutation is one of the most classical combinatorial structures. According to the linear representation of linked partitions, Chen $et$ $al$. (Chen W Y C, Wu S Y J, Yan C H. Linked partitions and linked cycles. $European$ $J$. $Combin$., 2008, 29(6): 1408–1426) defined the concept of singly covered minimal elements. Let $L(n,\,k)$ denote the set of linked partitions of $[n]$ with $k$ singly covered minimal elements and let $P(n,\,k)$ denote the set of permutations of $[n]$ with $k$ cycles. In this paper, we mainly establish two bijections between $L(n,\,k)$ and $P(n,\,k)$. The two bijections from a different perspective show the one-to-one correspondence between the singly covered minimal elements in $L(n,\,k)$ and the cycles in $P(n,\,k)$. 

  • Keywords

singly covered minimal element, linked partition, permutation, cycle

  • AMS Subject Headings

05A05, 05A18

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

guowkmath@163.com (Wenkui Guo)

  • BibTex
  • RIS
  • TXT
@Article{CMR-35-75, author = {Wenkui and Guo and guowkmath@163.com and 5911 and School of Science, Beijing Technology and Business University, Beijing, 100048 and Wenkui Guo and Feifei and Niu and and 5912 and School of Science, Beijing Technology and Business University, Beijing, 100048 and Feifei Niu}, title = {Singly Covered Minimal Elements of Linked Partitions and Cycles of Permutations}, journal = {Communications in Mathematical Research }, year = {2019}, volume = {35}, number = {1}, pages = {75--80}, abstract = {

Linked partitions were introduced by Dykema (Dykema K J. Multilinear function series and transforms in free probability theory. $Adv$. $Math$., 2005, 208(1): 351–407) in the study of the unsymmetrized T-transform in free probability theory. Permutation is one of the most classical combinatorial structures. According to the linear representation of linked partitions, Chen $et$ $al$. (Chen W Y C, Wu S Y J, Yan C H. Linked partitions and linked cycles. $European$ $J$. $Combin$., 2008, 29(6): 1408–1426) defined the concept of singly covered minimal elements. Let $L(n,\,k)$ denote the set of linked partitions of $[n]$ with $k$ singly covered minimal elements and let $P(n,\,k)$ denote the set of permutations of $[n]$ with $k$ cycles. In this paper, we mainly establish two bijections between $L(n,\,k)$ and $P(n,\,k)$. The two bijections from a different perspective show the one-to-one correspondence between the singly covered minimal elements in $L(n,\,k)$ and the cycles in $P(n,\,k)$. 

}, issn = {2707-8523}, doi = {https://doi.org/10.13447/j.1674-5647.2019.01.08}, url = {http://global-sci.org/intro/article_detail/cmr/13476.html} }
TY - JOUR T1 - Singly Covered Minimal Elements of Linked Partitions and Cycles of Permutations AU - Guo , Wenkui AU - Niu , Feifei JO - Communications in Mathematical Research VL - 1 SP - 75 EP - 80 PY - 2019 DA - 2019/12 SN - 35 DO - http://doi.org/10.13447/j.1674-5647.2019.01.08 UR - https://global-sci.org/intro/article_detail/cmr/13476.html KW - singly covered minimal element, linked partition, permutation, cycle AB -

Linked partitions were introduced by Dykema (Dykema K J. Multilinear function series and transforms in free probability theory. $Adv$. $Math$., 2005, 208(1): 351–407) in the study of the unsymmetrized T-transform in free probability theory. Permutation is one of the most classical combinatorial structures. According to the linear representation of linked partitions, Chen $et$ $al$. (Chen W Y C, Wu S Y J, Yan C H. Linked partitions and linked cycles. $European$ $J$. $Combin$., 2008, 29(6): 1408–1426) defined the concept of singly covered minimal elements. Let $L(n,\,k)$ denote the set of linked partitions of $[n]$ with $k$ singly covered minimal elements and let $P(n,\,k)$ denote the set of permutations of $[n]$ with $k$ cycles. In this paper, we mainly establish two bijections between $L(n,\,k)$ and $P(n,\,k)$. The two bijections from a different perspective show the one-to-one correspondence between the singly covered minimal elements in $L(n,\,k)$ and the cycles in $P(n,\,k)$. 

Wen-kui Guo & Fei-fei Niu. (2019). Singly Covered Minimal Elements of Linked Partitions and Cycles of Permutations. Communications in Mathematical Research . 35 (1). 75-80. doi:10.13447/j.1674-5647.2019.01.08
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