Continuous partition lattice

On Realization of Björner's 'continuous Partition Lattice' by Measurable Partitions

Björner [1] showed how a construction by von Neumann of examples of continuous geometries can be adapted to construct a continuous analogue of finite partition lattices. Björner's construction realizes the continuous partition lattice abstractly, as a completion of a direct limit of finite lattices. Here we give an alternative construction realizing a continuous partition lattice concretely as ...

Chains in the Noncrossing Partition Lattice

We establish recursions counting various classes of chains in the noncrossing partition lattice of a finite Coxeter group. The recursions specialize a general relation which is proven uniformly (i.e. without appealing to the classification of finite Coxeter groups) using basic facts about noncrossing partitions. We solve these recursions for each finite Coxeter group in the classification. Amon...

Symmetries of the k - bounded partition lattice

We generalize the symmetry on Young’s lattice, found by Suter, to a symmetry on the k-bounded partition lattice of Lapointe, Lascoux and Morse. Résumé. Nous généralisons la symmetrie sur le treillis de Young, découvert par Suter, à une symétrie sur le treillis des partages bornés par k et étudié par Lapointe, Lascoux and Morse.

Counting complements in the partition lattice, and hypertrees

A partition n = {A,, . . . . A,) of the set [n] = { 1, . . . . n} is an (unordered) family of nonempty subsets A,, . . . . A, of [n] which are pairwise disjoint and whose union is [n]. We call the Ai the blocks of rc, and let 1x1 =m. A partition {B,, . . . . B,} is a refinement of {A,, . . . . A,} if each Bj lies in some Ai. It is well known (but of no relevance in this paper) that the ordering...

The Size of the Largest Antichain in the Partition Lattice