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- W3025266871 abstract "A chord diagram E is a set of chords of a circle such that no pair of chords has a common endvertex. Let v 1 , v 2 , …, v 2 n be a sequence of vertices arranged in clockwise order along a circumference. A chord diagram { v 1 v n + 1 , v 2 v n + 2 , …, v n v 2 n } is called an n -crossing and a chord diagram { v 1 v 2 , v 3 v 4 , …, v 2 n − 1 v 2 n } is called an n -necklace. For a chord diagram E having a 2 -crossing S = { x 1 x 3 , x 2 x 4 } , the expansion of E with respect to S is to replace E with E 1 = ( E S ) ∪ { x 2 x 3 , x 4 x 1 } or E 2 = ( E S ) ∪ { x 1 x 2 , x 3 x 4 } . Beginning from a given chord diagram E as the root, by iterating chord expansions in both ways, we have a binary tree whose all leaves are nonintersecting chord diagrams. Let NCD ( E ) be the multiset of the leaves. In this paper, the multiplicity of an n -necklace in NCD ( E ) is studied. Among other results, it is shown that the multiplicity of an n -necklace generated from an n -crossing equals the Genocchi number when n is odd and the median Genocchi number when n is even." @default.
- W3025266871 created "2020-05-21" @default.
- W3025266871 creator A5087581400 @default.
- W3025266871 date "2020-10-24" @default.
- W3025266871 modified "2023-10-16" @default.
- W3025266871 title "The expansion of a chord diagram and the Genocchi numbers" @default.
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- W3025266871 doi "https://doi.org/10.26493/1855-3974.2239.7f1" @default.
- W3025266871 hasPublicationYear "2020" @default.
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