Year: 2019
Author: Wenkui Guo, Feifei Niu
Communications in Mathematical Research , Vol. 35 (2019), Iss. 1 : pp. 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 $P(n,\,k)$.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.13447/j.1674-5647.2019.01.08
Communications in Mathematical Research , Vol. 35 (2019), Iss. 1 : pp. 75–80
Published online: 2019-01
AMS Subject Headings: Global Science Press
Copyright: COPYRIGHT: © Global Science Press
Pages: 6
Keywords: singly covered minimal element linked partition permutation cycle