Linked partitions and linked cycles |
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Authors: | William YC Chen Susan YJ Wu Catherine H Yan |
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Institution: | aCenter for Combinatorics, LPMC-TJKLC, Nankai University, Tianjin 300071, PR China;bDepartment of Mathematics, Texas A&M University, College Station, TX 77843, United States |
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Abstract: | The notion of noncrossing linked partition arose from the study of certain transforms in free probability theory. It is known that the number of noncrossing linked partitions of n+1] is equal to the n-th large Schröder number rn, which counts the number of Schröder paths. In this paper we give a bijective proof of this result. Then we introduce the structures of linked partitions and linked cycles. We present various combinatorial properties of noncrossing linked partitions, linked partitions, and linked cycles, and connect them to other combinatorial structures and results, including increasing trees, partial matchings, k-Stirling numbers of the second kind, and the symmetry between crossings and nestings over certain linear graphs. |
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