Extensions of set partitions

Weak Compactness of the Set of ε-Extensions

Set Partitions in R

This short paper introduces a code snippet in the form of an R function that enumerates all possible partitions of a finite set given the sizes of the equivalence classes. Three combinatorial problems are solved using the software: one from bioinformatics, one from scheduling theory, and one from forensic science.

Records in Set Partitions

A partition of [n] = {1, 2, . . . , n} is a decomposition of [n] into nonempty subsets called blocks. We will make use of the canonical representation of a partition as a word over a finite alphabet, known as a restricted growth function. An element ai in such a word π is a strong (weak) record if ai > aj (ai > aj) for all j = 1, 2, . . . , i−1. Furthermore, the position of this record is i. We...

Front Representation of Set Partitions

Let π be a set partition of [n] = {1, 2, . . . , n}. The standard representation of π is the graph on the vertex set [n] whose edges are the pairs (i, j) of integers with i < j in the same block which does not contain any integer between i and j. The front representation of π is the graph on the vertex set [n] whose edges are the pairs (i, j) of integers with i < j in the same block whose small...

Linear extensions and order-preserving poset partitions

We examine the lattice of all order congruences of a finite poset from the viewpoint of combinatorial algebraic topology. We will prove that the order complex of the lattice of all nontrivial order congruences (or order-preserving partitions) of a finite n-element poset P with n ≥ 3 is homotopy equivalent to a wedge of spheres of dimension n − 3. If P is connected, then the number of spheres is...