Singly Covered Minimal Elements of Linked Partitions and Cycles of Permutations (Communications in Mathematical Research, 2020)
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 $P(n,\,k)$.