Primes, irreducibles and extremal lattices

The KP-II equation possesses a class of line soliton solutions which can be qualitatively described via a tropical approximation as a chain of rooted binary trees, except at "critical" events where a transition to a different rooted binary tree takes place. We prove that these correspond to maximal chains in Tamari lattices (which are poset structures on associahedra). We further derive results that allow to compute details of the evolution, including the critical events. Moreover, we present some insights into the structure of the more general line soliton solutions. All this yields a characterization of possible evolutions of line soliton patterns on a shallow fluid surface (provided that the KP-II approximation applies).

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“…Hence the chain is maximal. The total number of a's is r − 1, which is known to be the shortest length of a maximal Tamari chain [54].…”

Section: A1 Preparationsmentioning

confidence: 99%

KP line solitons and Tamari lattices

Dimakis¹,

Müller‐Hoissen²

2010

J. Phys. A: Math. Theor.

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“…Proof. T n has a chain of length n 2 and this is the maximum length [Mar92]. In T n (m) , consider the chain whose vertex x i is the (m, n)-Dyck partition λ = (λ 1 , .…”

Section: Maximum Length Chains In T Nmentioning

confidence: 99%

Chains of maximum length in the Tamari lattice

Fishel

Nelson

2014

Proc. Amer. Math. Soc.

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Abstract. The Tamari lattice Tn was originally defined on bracketings of a set of n+1 objects, with a cover relation based on the associativity rule in one direction. Although in several related lattices, the number of maximal chains is known, quoting Knuth, "The enumeration of such paths in Tamari lattices remains mysterious."The lengths of maximal chains vary over a great range. In this paper, we focus on the chains with maximum length in these lattices. We establish a bijection between the maximum length chains in the Tamari lattice and the set of standard shifted tableaux of staircase shape. We thus derive an explicit formula for the number of maximum length chains, using the Thrall formula for the number of shifted tableaux. We describe the relationship between chains of maximum length in the Tamari lattice and certain maximal chains in weak Bruhat order on the symmetric group, using standard Young tableaux. Additionally, recently, Bergeron and Préville-Ratelle introduced a generalized Tamari lattice. Some of the results mentioned above carry over to their generalized Tamari lattice.

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“…This is one way to construct the Tamari lattice, which was introduced in [20] and which has since been studied by several authors (see [9,13,8,12,4]). Although this is not clear from the elementary description given here, the Tamari lattice should be thought of as belonging to type A.…”

Section: Introductionmentioning

confidence: 99%

Tamari lattices and noncrossing partitions in type B

Thomas

2006

Discrete Mathematics

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The usual, or type A n , Tamari lattice is a partial order on T A n , the triangulations of an (n + 3)-gon. We define a partial order on T B n , the set of centrally symmetric triangulations of a (2n + 2)-gon. We show that it is a lattice, and that it shares certain other nice properties of the A n Tamari lattice, and therefore that it deserves to be considered the B n Tamari lattice. We also define a bijection between T B n and the noncrossing partitions of type B n defined by Reiner.

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Primes, irreducibles and extremal lattices

Cited by 59 publications

References 17 publications

KP line solitons and Tamari lattices

KP line solitons and Tamari lattices

Chains of maximum length in the Tamari lattice

Tamari lattices and noncrossing partitions in type B

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