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\(k\)-distant crossings and nestings of matchings and partitions. (English. French summary) Zbl 1391.05044

Krattenthaler, Christian (ed.) et al., Proceedings of the 21st annual international conference on formal power series and algebraic combinatorics, FPSAC 2009, Hagenberg, Austria, July 20–24, 2009. Nancy: The Association. Discrete Mathematics & Theoretical Computer Science (DMTCS). Discrete Mathematics and Theoretical Computer Science. Proceedings, 349-360 (2009).
Summary: We define and consider \(k\)-distant crossings and nestings for matchings and set partitions, which are a variation of crossings and nestings in which the distance between vertices is important. By modifying an involution of A. Kasraoui and J. Zeng [Electron. J. Comb. 13, No. 1, Research paper R33, 12 p. (2006; Zbl 1096.05006)], we show that the joint distribution of \(k\)-distant crossings and nestings is symmetric. We also study the numbers of \(k\)-distant noncrossing matchings and partitions for small \(k\), which are counted by well-known sequences, as well as the orthogonal polynomials related to \(k\)-distant noncrossing matchings and partitions. We extend W. Y. C. Chen et al.’s [Trans. Am. Math. Soc. 359, No. 4, 1555–1575 (2007; Zbl 1108.05012)] \(r\)-crossings and enhanced \(r\)-crossings.
For the entire collection see [Zbl 1196.05001].

MSC:

05A18 Partitions of sets
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
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