## Crossings and nestings of matchings and partitions

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Venue: | Trans. Amer. Math. Soc |

Citations: | 57 - 15 self |

### BibTeX

@ARTICLE{Chen_crossingsand,

author = {William Y. C. Chen and Eva Y. P. Deng and Rosena R. X. Du and Richard P. Stanley and Catherine and H. Yan},

title = {Crossings and nestings of matchings and partitions},

journal = {Trans. Amer. Math. Soc},

year = {},

volume = {359},

pages = {2007}

}

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### Abstract

Abstract. 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]. As a corollary, the number of knoncrossing partitions is equal to the number of k-nonnesting partitions. The same is also true for matchings. An application is given to the enumeration of matchings with no k-crossing (or with no k-nesting). 1.

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Citation Context ...m − 1, ..., 1) of equation (38) in [12], giving (5.1) where Fk(x) = det[Ii−j(2x) − Ii+j(2x)] k−1 i,j=1 , Im(2x) = � j≥0 x m+2j j!(m + j)! , the hyperbolic Bessel function of the first kind of order m =-=[30]-=-. One can easily check that when k = 2, the generating function of 2-noncrossing matchings equals F2(x) = I0(2x) − I2(2x) = � j≥0 Cj x2j (2j)! ,sCROSSINGS AND NESTINGS 17 where Cj is the j-th Catalan ... |

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Citation Context ... of arcs connecting the elements of each block in numerical order. Such an edge set is called the standard representation of the partition P. For example, the standard representation of 1457-26-3 is {=-=(1, 4)-=-, (4, 5), (5, 7), (2, 6)}. Here we always write an arc e as a pair (i, j) with i<j, and say that i is the left-hand endpoint of e and j is the right-hand endpoint of e. Let k ≥ 2andP∈ Πn. Define a k-c... |

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Citation Context ...elements of each block in numerical order. Such an edge set is called the standard representation of the partition P. For example, the standard representation of 1457-26-3 is {(1, 4), (4, 5), (5, 7), =-=(2, 6)-=-}. Here we always write an arc e as a pair (i, j) with i<j, and say that i is the left-hand endpoint of e and j is the right-hand endpoint of e. Let k ≥ 2andP∈ Πn. Define a k-crossing of P as a k-subs... |

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Citation Context ...the duration of the game in the classical ruin problem, that is, restricted random walks with absorbing barriers at 0 and a, and initial position z. See, for example, Equation (4.11) of Chapter 14 of =-=[9]-=-: let ∞� Uz(x)= uz,mx m , m=0s1572 W. CHEN ET AL. where uz,n is the probability that the process ends with the n-thstepatthebarrier 0. Then � �z a−z q λ1 (x) − λ Uz(x)= p a−z 2 (x) λa 1 (x) − λa2 (x) ... |

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Citation Context ...on {(x + 1, ∗)}, and again we have A(u1 · · · usv1 · · · vt−1vt) = A(u1 · · · usjv1 · · · vt−1vt) \ {j}. This finishes the proof of the claim. Step 4. We shall need the following theorem of Schensted =-=[22]-=-[25, Thms. 7.23.13, 7.23.17], which gives the basic connection between the RSK algorithm and the increasing and decreasing subsequences. Schensted’s Theorem Let σ be a sequence of integers whose terms... |

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Citation Context ... − 1, ..., 1) of equation (38) in [12], giving (5.1) where Fk(x) = det [Ii−j(2x) − Ii+j(2x)] k−1 i,j=1 , Im(2x) = ∑ j≥0 x m+2j j!(m + j)! , the hyperbolic Bessel function of the first kind of order m =-=[30]-=-. One can easily check that when k = 2, the generating function of 2-noncrossing matchings equals F2(x) = I0(2x) − I2(2x) = ∑ j≥0 Cj x2j (2j)! ,CROSSINGS AND NESTINGS 17 where Cj is the j-th Catalan ... |

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Citation Context ...s y = 0 and y = k. The evaluation of gk,1(m) was first considered by Takács in [28] by a probabilistic argument. Explicit formula and generating function for this case are well-known. For example, in =-=[19]-=- one obtains the explicit formula by applying the reflection principle repeatedly, viz., gk,1(m) = � �� � � �� 2m 2m − . m − i(k + 2) m + i(k + 2) + k + 1 The generating function Gk,1(x) is a special ... |

69 | Algebraic aspects of increasing subsequences
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(Show Context)
Citation Context ...crossing Dyck paths due to Gouyou-Beauchamps in [11]. Remark 5.6. The determinant formula (5.1) has been studied by Baik and Rains in [2, Eqs. (2.25)]. One simply puts i−1forjand j −1forkin (2.25) of =-=[2]-=- to get our formula. The same formula was also obtained by Goulden [10] as the generating function for fixed-point-free permutations with no decreasing subsequence of length greater than 2k. See Theor... |

62 |
An extension of Schensted’s theorem
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Citation Context ...sequencew of distinct integers. Schensted’s Theorem provides a combinatorial interpretation of the terms λ1 and λ ′ 1:theyare the length of the longest increasing and decreasing subsequences of w. In =-=[14]-=- C. Greene extended Schensted’s Theorem by giving an interpretation of the rest of the diagram of λ =(λ1,λ2,...). Assume w is a sequence of length n. Foreachk ≤ n,letdk(w) denote the length of the lon... |

52 |
A combinatorial problem connected with differential equations
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(Show Context)
Citation Context ...and endpoint of another arc (j, k), then delete j first, and then insert i. The vacillating tableau φ(P) is the sequence of shapes of the above SYT’s. Example 3.1. Let P be the partition 1457-26-3 of =-=[7]-=-. � � � � � � � 1 2 3 4 5 6 7 Figure 1. The standard representation of the partition 1457-26-3. Starting from ∅ on the right, go from 7 to 1. The seven steps are: (1) do nothing, then insert 5, (2) do... |

47 | The asymptotics of monotone subsequences of involutions
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(Show Context)
Citation Context ...1.1 and 2.3 of [10] and specialize hi to be xi /i!, so gl becomes the hyperbolic Bessel function. The asymptotic distribution of cr(M) follows from another result of Baik and Rains. In Theorem 3.1 of =-=[3]-=- they obtained the limit distribution for the length of the longest decreasing subsequence of fixedpoint-free involutions w. But representing w as a matching M, the condition that w has no decreasing ... |

46 |
Longest increasing and decreasing subsequences, Canad
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(Show Context)
Citation Context ...the last position {(x + 1, ∗)}, and again we have A(u1 ···usv1 ···vt−1vt)=A(u1 ···usjv1 ···vt−1vt) \{j}. This finishes the proof of the claim. Step 4. We shall need the following theorem of Schensted =-=[22]-=-, [25, Thms. 7.23.13, 7.23.17], which gives the basic connection between the RSK algorithm and the increasing and decreasing subsequences. Schensted’s Theorem. Let σ be a sequence of integers whose te... |

42 | Enumerative Combinatorics, vol. 1, Wadsworth and Brooks/Cole - Stanley - 1986 |

40 |
Sur un probleme de configuration et sur les fractions continues
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- 1952
(Show Context)
Citation Context ...ings of matchings has been studied for the cases k =2andk =3. Fork = 2, in addition to the above results on Catalan numbers, the distribution of the number of 2-crossings has been studied by Touchard =-=[29]-=-, and later more explicitly by Riordan [21], who gave a generating function. M. de SainteCatherine [8] proved that 2-crossings and 2-nestings are identically distributed over all matchings of [2n], i.... |

38 |
Differential posets
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Citation Context ...square), and similarly D takes a shape to the sum of all shapes that it covers in Young’s lattice (i.e., by deleting a square). Then, as is well known, DU − UD = I (the identity operator). See, e.g., =-=[23]-=-, [25, Exer. 7.24]. It follows that (4.2) (U + I)(D + I)=DU + ID + UI. Iterating the left-hand side generates vacillating tableaux, and iterating the righthand side gives the hesitating tableaux defin... |

36 | Random walks in Weyl chambers and the decomposition of tensor products
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(Show Context)
Citation Context ...the region Vk from the origin to itself with step set {±ɛ1, ±ɛ2,...,±ɛk−1}. Set Fk(x)= � m fk(m) x2m (2m)! . It turns out that a determining expression for Fk(x) has been given by Grabiner and Magyar =-=[12]-=-. It is simply the case λ = η =(m, m − 1, ..., 1) of equation (38) in [12], giving (5.1) where Fk(x)=det[Ii−j(2x) − Ii+j(2x)] k−1 i,j=1 , Im(2x)= � j≥0 x m+2j j!(m + j)! , the hyperbolic Bessel functi... |

35 | Partition algebras
- Halverson, Ram
(Show Context)
Citation Context ...ebra Pn is a certain semisimple algebra, say over C, whose dimension is the Bell number B(n) (the number of partitions of [n]). (The algebra Pn depends on a parameter x which is irrelevant here.) See =-=[15, 16]-=- for a surveysCROSSINGS AND NESTINGS 1561 of this topic. Vacillating tableaux are related to irreducible representations of Pn in the same way that SYT of content [n] are related to irreducible repres... |

29 |
The Cauchy identity for Sp(2n
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- 1990
(Show Context)
Citation Context ...get a bijection between complete matchings on [2m] and oscillating tableaux of empty shape and length 2m. This bijection was originally constructed by the fourth author, and then extended by Sundaram =-=[27]-=- to arbitrary shapes to give a combinatorial proof of the Cauchy identity for the symplectic group Sp(2m). The explicit description of the bijection has appeared in [27] and was included in [25, Exerc... |

26 |
A Schensted-type correspondence for the symplectic group
- Berele
- 1986
(Show Context)
Citation Context ...the symplectic group Sp(2m). The explicit description of the bijection has appeared in [27] and was included in [25, Exercise 7.24]. Oscillating tableaux first appeared (though not with that name) in =-=[5]-=-. Recall that the ordinary RSK algorithm gives a bijection between the symmetric group Sm and pairs (P,Q) of SYTs of the same shape λ ⊢ m. This result and Schensted’s Theorem can be viewed as a specia... |

22 |
The distribution of crossing chords joining pairs of 2n points on a circle
- Riordan
(Show Context)
Citation Context ...cases k =2andk =3. Fork = 2, in addition to the above results on Catalan numbers, the distribution of the number of 2-crossings has been studied by Touchard [29], and later more explicitly by Riordan =-=[21]-=-, who gave a generating function. M. de SainteCatherine [8] proved that 2-crossings and 2-nestings are identically distributed over all matchings of [2n], i.e., the number of matchings with r 2-crossi... |

19 |
Sainte-Catherine, Couplages et Pfaffiens en combinatoire, physique et informatique
- de
- 1983
(Show Context)
Citation Context ...s on Catalan numbers, the distribution of the number of 2-crossings has been studied by Touchard [29], and later more explicitly by Riordan [21], who gave a generating function. M. de SainteCatherine =-=[8]-=- proved that 2-crossings and 2-nestings are identically distributed over all matchings of [2n], i.e., the number of matchings with r 2-crossings is equal to the number of matchings with r 2-nestings. ... |

16 | Random walk in an alcove of an affine Weyl group, and non-colliding random walks on an interval - Grabiner |

14 |
their relatives, and algebraic differential equatsion
- Klazar, numbers
- 2003
(Show Context)
Citation Context ...≤r≤ t. Remark 3.7. In our definition, a k-crossing is defined as a set of k mutually crossing arcs in the standard representation of the partition. There exist some other definitions. For example, in =-=[17]-=- M. Klazar defined the 3-noncrossing partition as a partition P which does not have 3 mutually crossing blocks. It can be seen that P is 3-noncrossing in Klazar’s sense if and only if there do not exi... |

12 | A map-theoretic approach to Davenport-Schinzel sequences - Mullin, Stanton - 1972 |

11 | Combinatorial representation theory
- Barcelo, Ram
- 1999
(Show Context)
Citation Context ... of arcs connecting the elements of each block in numerical order. Such an edge set is called the standard representation of the partition P. For example, the standard representation of 1457-26-3 is {=-=(1, 4)-=-, (4, 5), (5, 7), (2, 6)}. Here we always write an arc e as a pair (i, j) with i<j, and say that i is the left-hand endpoint of e and j is the right-hand endpoint of e. Let k ≥ 2andP∈ Πn. Define a k-c... |

11 | RSK insertion for set partitions and diagram algebras
- Halverson, Lewandowski
(Show Context)
Citation Context ...en followed by the reverse of a walk from ∅ to λ in m steps. It follows that � (2.3) gλ(n)gλ(m)=g∅(m + n)=B(m + n). λ For the case m = n = k, the identity (2.3) is proved by Halverson and Lowandowski =-=[16]-=-, who gave a bijective proof using similar procedures as those in ψ. (3) The partition algebra Pn is a certain semisimple algebra, say over C, whose dimension is the Bell number B(n) (the number of pa... |

9 |
A linear operator for symmetric functions and tableaux in a strip with given trace
- Goulden
- 1992
(Show Context)
Citation Context ...inant formula (5.1) has been studied by Baik and Rains in [2, Eqs. (2.25)]. One simply puts i−1 for j and j −1 for k in (2.25) of [2] to get our formula. The same formula was also obtained by Goulden =-=[10]-=- as the generating function for fixed point free permutations with no decreasing subsequence of length greater than 2k. See Theorem 1.1 and 2.3 of [10] and specialize hi to be xi /i!, so gl becomes th... |

4 |
Standard Young tableaux of height 4 and 5
- Gouyou-Beauschamps
- 1989
(Show Context)
Citation Context ...uted over all matchings of [2n], i.e., the number of matchings with r 2-crossings is equal to the number of matchings with r 2-nestings. The enumeration of 3-nonnesting matchings was first studied in =-=[11]-=- by GouyouBeauschamps, where he gave a bijection between involutions with no decreasing sequence of length 6 and pairs of noncrossing Dyck left factors by a recursive construction. His bijection is es... |

2 |
Standard Young tableaux of height 4
- Gouyou-Beauschamps
- 1989
(Show Context)
Citation Context ...ngs of [2n], i.e., the number of matchings with r 2-crossings is equal to the number of matchings with r 2-nestings. The enumeration of 3-nonnesting matchings was first studied by Gouyou -Beauschamps =-=[11]-=-, in which he gave a bijection between involutions with no decreasing sequence of length 6 and pairs of noncrossing Dyck left factors by a recursive construction. His bijection is essentially a corres... |

2 | An extension of Schensted's theorem, Adv - Greene - 1974 |

1 |
Supplementary Exercises for Chapter 7 of Enumerative Combinatorics, available at http://www-math.mit.edu/,rstan/ec
- Stanley
- 1999
(Show Context)
Citation Context ...d item (ww) can be viewed as nonnesting matchings, in which the blocks of the matching are the columns of the standard Young tableaux of shape (n,n). Nonnesting matchings are also one of the items of =-=[26]-=-. Let k ≥ 2 be an integer. A k-crossing of a matching M is a set of k arcs (ir1 ,jr1 ), (ir2 ,jr2 ),...,(irk ,jrk )ofM such that ir1 <ir2 < ···<irk <jr1 <jr2 < ···<jrk . A matching without any k-cross... |

1 |
The Cauchy identity for Sp(2n),J.Combin.TheorySer.A,53
- Sundaram
- 1990
(Show Context)
Citation Context ...get a bijection between complete matchings on [2m] and oscillating tableaux of empty shape and length 2m. This bijection was originally constructed by the fourth author, and then extended by Sundaram =-=[27]-=- to arbitrary shapes to give a combinatorial proof of the Cauchy identity for the symplectic group Sp(2m). The explicit description of the bijection has appeared in [27] and was included in [25, Exerc... |

1 |
MR1424469 (97k:01072) Center for Combinatorics, LPMC, Nankai University, Tianjin 300071, People’s Republic of China E-mail address: chen@nankai.edu.cn
- Whittaker, Watson, et al.
- 1927
(Show Context)
Citation Context ...m, m − 1, ..., 1) of equation (38) in [12], giving (5.1) where Fk(x)=det[Ii−j(2x) − Ii+j(2x)] k−1 i,j=1 , Im(2x)= � j≥0 x m+2j j!(m + j)! , the hyperbolic Bessel function of the first kind of order m =-=[30]-=-. One can easily check that when k = 2, the generating function of 2-noncrossing matchings equals F2(x)=I0(2x) − I2(2x)= � x Cj 2j (2j)! , j≥0sCROSSINGS AND NESTINGS 1571 where Cj is the j-th Catalan ... |

1 |
Ballot problems
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- 1962
(Show Context)
Citation Context ...1, the number gk,1(m) counts lattice paths from (0, 0) to (2m, 0) with steps (1, 1) or (1, −1) that stay between the lines y = 0 and y = k. The evaluation of gk,1(m) was first considered by Takács in =-=[28]-=- by a probabilistic argument. Explicit formula and generating function for this case are well-known. For example, in [19] one obtains the explicit formula by applying the reflection principle repeated... |

1 |
Partition algebras, preprint; math.RT/0401314
- Halverson, Ram
(Show Context)
Citation Context ...ebra Pn is a certain semisimple algebra, say over C, whose dimension is the Bell number B(n) (the number of partitions of [n]). (The algebra Pn depends on a parameter x which is irrelevant here.) See =-=[13]-=- for a survey of this topic. Vacillating tableaux are related to irreducible representations of Pn in the same way that SYT of content [n] are related to irreducible 7representations of the symmetric... |

1 |
Schensted insertion for set partitions and the partition algebra
- Halverson, Lewandowski
- 2004
(Show Context)
Citation Context ...en followed by the reverse of a walk from ∅ to λ in m steps. It follows that ∑ gλ(n)gλ(m) = g∅(m + n) = B(m + n). (6) λ For the case m = n = k, the identity (6) is proved by Halverson and Lowandowski =-=[15]-=-, who gave a bijective proof using similar procedures as those in ψ. (3). The partition algebra Pn is a certain semisimple algebra, say over C, whose dimension is the Bell number B(n) (the number of p... |

1 |
AND NESTINGS 21 Center for Combinatorics, LPMC, Nankai University, Tianjin 300071, P.R. China E-mail address: chen@nankai.edu.cn
- Whittaker, Watson, et al.
- 1927
(Show Context)
Citation Context ... m − 1, ..., 1) of equation (38) in [12], giving (5.1) where Fk(x) =det[Ii−j(2x) − Ii+j(2x)] k−1 i,j=1 , Im(2x) = ∑ j≥0 x m+2j j!(m + j)! , the hyperbolic Bessel function of the first kind of order m =-=[30]-=-. One can easily check that when k = 2, the generating function of 2-noncrossing matchings equals F2(x) =I0(2x) − I2(2x) = ∑ x Cj 2j (2j)! , j≥0CROSSINGS AND NESTINGS 17 where Cj is the j-th Catalan ... |

1 | Gebiete 1 - Verw - 1962 |