# Crossings and nestings of matchings and partitions

@article{Chen2005CrossingsAN,
title={Crossings and nestings of matchings and partitions},
author={William Y. C. Chen and Eva Yu-Ping Deng and Rosena R. X. Du and Richard P. Stanley and Catherine Huafei Yan},
journal={Transactions of the American Mathematical Society},
year={2005},
volume={359},
pages={1555-1575}
}
• William Y. C. Chen, +2 authors C. Yan
• Published 14 January 2005
• Mathematics
• Transactions of the American Mathematical Society
We present results on the enumeration of crossings and nestings for matchings and set partitions. Using a bijection between partitions and vacillating tableaux, we show that if we fix the sets of minimal block elements and maximal block elements, the crossing number and the nesting number of partitions have a symmetric joint distribution. It follows that the crossing numbers and the nesting numbers are distributed symmetrically over all partitions of [n], as well as over all matchings on [2n… Expand
212 Citations

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#### References

SHOWING 1-10 OF 54 REFERENCES
Random Walks in Weyl Chambers and the Decomposition of Tensor Powers
• Mathematics
• 1993
We consider a class of random walks on a lattice, introduced by Gessel and Zeilberger, for which the reflection principle can be used to count the number of k-step walks between two points which stayExpand
An Extension of Schensted's Theorem
Publisher Summary This chapter presents an extension of Schensted's theorem. σ = ( a 1 , a 2 , …, a n ) is a sequence of integers whose terms are distinct. The chapter presents a theorem that statesExpand
Random Walk in an Alcove of an Affine Weyl Group, and Non-colliding Random Walks on an Interval
We use a reflection argument, introduced by Gessel and Zeilberger, to count the number of k-step walks between two points which stay within a chamber of a Weyl group. We apply this technique to walksExpand
The asymptotics of monotone subsequences of involutions
• Mathematics, Physics
• 1999
We compute the limiting distributions of the lengths of the longest monotone subsequences of random (signed) involutions with or without conditions on the number of fixed points (and negated points)Expand
Longest Increasing and Decreasing Subsequences
This paper deals with finite sequences of integers. Typical of the problems we shall treat is the determination of the number of sequences of length n, consisting of the integers 1,2, ... , m, whichExpand
The distribution of crossings of chords joining pairs of $2n$ points on a circle
This paper, in the first place, calls attention to an extraordinarily compact solution of the problem in the title, given (a trifle hidden) in the work of the late Jacques Touchard. Its main weight,Expand
Bell numbers, their relatives, and algebraic differential equations
• M. Klazar
• Computer Science, Mathematics
• J. Comb. Theory, Ser. A
• 2003
It is proved that the ordinary generating function of Bell numbers satisfies no algebraic differential equation over C(x) (in fact, over a larger field) and related numbers counting various set partitions are investigated. Expand
RSK Insertion for Set Partitions and Diagram Algebras
• Mathematics, Computer Science
• Electron. J. Comb.
• 2005
This work gives combinatorial proofs of two identities from the representation theory of the partition algebra of C A_k(n), n = 2k, where n is the number of standard tableaux of shape $\lambda, and m_k^\lambda is theNumber of "vacillating tableaux" of shape$2k and length 2k. Expand
Standard Young Tableaux of Height 4 and 5
This work gives exact formulas for the number of standard Young tableaux having n cells and at most k rows in the cases k = 4 and k = 5 and deduces that the corresponding generating functions are not algebraic. Expand
What Is Enumerative Combinatorics
The basic problem of enumerative combinatorics is that of counting the number of elements of a finite set. Usually are given an infinite class of finite sets S i where i ranges over some index set IExpand