✦ LIBER ✦

On the structure of the lattice of noncrossing partitions

✍ Scribed by Rodica Simion; Daniel Ullman

Publisher: Elsevier Science
Year: 1991
Tongue: English
Weight: 882 KB
Volume: 98
Category: Article
ISSN: 0012-365X
DOI: 10.1016/0012-365x(91)90376-d

No coin nor oath required. For personal study only.

✦ Synopsis

Simion, R. and D. Ullman, On the structure of the lattice of noncrossing partitions, Discrete Mathematics 98 (1991) 193-206.

We show that the lattice of noncrossing (set) partitions is self-dual and that it admits a symmetric chain decomposition.

The self-duality is proved via an order-reversing involution. Two proofs are given of the existence of the symmetric chain decomposition, one recursive and one constructive.

Several identities involving Catalan numbers emerge from the construction of the symmetric chain decomposition.

📜 SIMILAR VOLUMES

Chains in the lattice of noncrossing par

Chains in the lattice of noncrossing partitions

✍ Paul H. Edelman; Rodica Simion 📂 Article 📅 1994 🏛 Elsevier Science 🌐 English ⚖ 735 KB

The lattice of noncrossing set partitions is known to admit an R-labeling. Under this labeling, maximal chains give rise to permutations. We discuss structural and enumerative properties of the lattice of noncrossing partitions, which pertain to a new permutation statistic, m(a), defined as the num

Representations of a sublattice of the p

Representations of a sublattice of the partition lattice on a lattice

✍ Ronny Rousseau 📂 Article 📅 1983 🏛 Elsevier Science 🌐 English ⚖ 529 KB

A partial ordering is defined for monotone projections on a lattice, such that the set of these mappings is a lattice which is isomorphic to a sublattice of the partition lattice.

The Lattice of Non-crossing Partitions a

The Lattice of Non-crossing Partitions and the Birkhoff-Lewis Equations

✍ D.M. Jackson 📂 Article 📅 1994 🏛 Elsevier Science 🌐 English ⚖ 186 KB

A matrix associated with the chromatic join of non-crossing partitions has been introduced by Tutte to generalise the Birkhoff-Lewis equations. A conjecturc is given for its determinant in terms of polynomials having the Beraha numbers among their roots. Corrcsponding results for join and meet on th

The Size of the Largest Antichain in the

The Size of the Largest Antichain in the Partition Lattice

✍ E.Rodney Canfield 📂 Article 📅 1998 🏛 Elsevier Science 🌐 English ⚖ 268 KB

Consider the poset 6 n of partitions of an n-element set, ordered by refinement. The sizes of the various ranks within this poset are the Stirling numbers of the second kind. Let a= 1 2 &e log(2)Â4. We prove the following upper bound for the ratio of the size of the largest antichain to the size of

Partition Function of the Eight-Vertex L

Partition Function of the Eight-Vertex Lattice Model

✍ Rodney J. Baxter 📂 Article 📅 2000 🏛 Elsevier Science 🌐 English ⚖ 269 KB

The partition function of the zero-field ``Eight-Vertex'' model on a square M by N lattice is calculated exactly in the limit of M, N large. This model includes the dimer, ice and zerofield Ising, F and KDP models as special cases. In general the free energy has a branch point singularity at a phase

The Structure of the Partition Algebras

✍ Paul Martin 📂 Article 📅 1996 🏛 Elsevier Science 🌐 English ⚖ 353 KB

Ä 4 1 1 2 2 3 3 4 4 5 5 Ž ÄÄ 4 as clusters, and of composition of partitions ab s Q. ␣ , ␣ , ␣ , ␣ , 1 2 3 4 Ä 4 Ä 4 Ä 4 4 . Ž ␣ , ␤ , ␤ , ␤ , ␤ , ␤ by an appropriate juxtaposition cf. p. 868 5 1 2 3 4 5 w x. of 2 . We define the elements of S , Ä␣ , ␤ 4 n Ä 4 Ä 4 Ä 4 Ä 4