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 - <p style="text-align: justify;">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 $L(n,\,k)$&nbsp;and the cycles in $P(n,\,k)$.&nbsp;</p>