The Cauchy identity for Sp(2n)

Sundaram, Sheila

doi:10.1016/0097-3165(90)90058-5

Cited by 73 publications

(78 citation statements)

References 5 publications

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“…Thus we 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,Exercise 7.24].…”

Section: A Variant: Partitions and Hesitating Tableauxmentioning

confidence: 99%

“…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,Exercise 7.24]. Oscillating tableaux first appeared (though not with that name) in [5].…”

Section: A Variant: Partitions and Hesitating Tableauxmentioning

confidence: 99%

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Crossings and nestings of matchings and partitions

Chen

Deng

et al. 2006

Trans. Amer. Math. Soc.

186

418

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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).

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Section: A Variant: Partitions and Hesitating Tableauxmentioning

confidence: 99%

Section: A Variant: Partitions and Hesitating Tableauxmentioning

confidence: 99%

Crossings and nestings of matchings and partitions

Chen

Deng

et al. 2006

Trans. Amer. Math. Soc.

186

418

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“…Theorem 15 is proved by using a version of RSK first defined by the author (unpublished) and then extended by Sundaram [103]. Define an oscillating tableau of shape λ n and length k to be a sequence…”

Section: Theorem 15 For All I J N We Have F N (I J ) = F N (J I)mentioning

confidence: 99%

Increasing and decreasing subsequences and their variants

Stanley¹

2007

Proceedings of the International Congress of Mathematicians Madrid, August 22–30, 2006

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Abstract. We survey the theory of increasing and decreasing subsequences of permutations. Enumeration problems in this area are closely related to the RSK algorithm. The asymptotic behavior of the expected value of the length is(w) of the longest increasing subsequence of a permutation w of 1, 2, . . . , n was obtained by Vershik-Kerov and (almost) by Logan-Shepp. The entire limiting distribution of is(w) was then determined by Baik, Deift, and Johansson. These techniques can be applied to other classes of permutations, such as involutions, and are related to the distribution of eigenvalues of elements of the classical groups. A number of generalizations and variations of increasing/decreasing subsequences are discussed, including the theory of pattern avoidance, unimodal and alternating subsequences, and crossings and nestings of matchings and set partitions. Mathematics Subject Classification (2000). Primary 05A05, 05A06; Secondary 60C05.

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“…In general these tableaux map onto certain two line arrays [22], which in the present case represent involutions of {1, 2, . .…”

mentioning

confidence: 99%

“…Following [22] we work backwards in the construction of the two line array from the sequence of oscillating tableaux. In going from the tableau at step i to that at step i − 1 there are two distinct situations.…”

mentioning

confidence: 99%

Random walks and random fixed-point free involutions

Baker

Forrester

2001

J. Phys. A: Math. Gen.

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A bijection is given between fixed point free involutions of {1, 2, . . . , 2N } with maximum decreasing subsequence size 2p and two classes of vicious (non-intersecting) random walker configurations confined to the half line lattice points l ≥ 1. In one class of walker configurations the maximum displacement of the right most walker is p. Because the scaled distribution of the maximum decreasing subsequence size is known to be in the soft edge GOE (random real symmetric matrices) universality class, the same holds true for the scaled distribution of the maximum displacement of the right most walker.

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The Cauchy identity for Sp(2n)

Cited by 73 publications

References 5 publications

Crossings and nestings of matchings and partitions

Crossings and nestings of matchings and partitions

Increasing and decreasing subsequences and their variants

Random walks and random fixed-point free involutions

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