(untitled) (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).