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

Wenkui Guo & Feifei Niu

DOI: 10.13447/j.1674-5647.2019.01.08

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 = {Guo , Wenkui and Niu , Feifei}, 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 $P(n,\,k)$. 

Guo , Wenkui and Niu , Feifei. (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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