If a1, a2, . . . , ak and n are positive integers such that n = a1+a2+· · ·+ak, then the sum a1 + a2 + · · ·+ ak is said to be a partition of n of length k, and a1, a2, . . . , ak are said to be the parts of the partition. Two partitions that differ only in the order of their parts are considered to be the same partition. Let Pn be the set of partitions of n, and let Pn,k be the set of partitio...

In 2009, Blagojević, Matschke & Ziegler established the first tight colored Tverberg theorem. We develop a colored version of our previous results (2008): Evenness and non-trivial lower bounds for the number of colored Tverberg partitions. Both properties follow from similar results on the number of colored Birch partitions.

Recently the authors have proposed tests for the one-sample and the ksample problem, and a test for independence. All three tests are based on sample space partitions, but they were originally developed in different papers. Here we give an overview of the construction of these tests, stressing the common underlying concept of “sample space partitions.”

In this paper a morphological contrast measure is introduced. The quantification of the contrast is based on the analysis of the edges, which are associated with substantial changes in luminance. Due to this, the contrast measure is used to detect the image that presents a high visual contrast when a set of output images is analyzed. The set of output images is obtained by application of morpho...

An M-partition of a positive integer m is a partition of m with as few parts as possible such that every positive integer less than m can be written as a sum of parts taken from the partition. This type of partition is a variation of MacMahon’s perfect partition, and was recently introduced and studied by O’Shea, who showed that for half the numbers m, the number of M-partitions of m is equal t...

We introduce partition equilibrium and study its existence in resource selection games (RSG). In partition equilibrium the agents are partitioned into coalitions, and only deviations by the prescribed coalitions are considered. This is in difference to the classical concept of strong equilibrium according to which any subset of the agents may deviate. In resource selection games, each agent sel...

Let P1, . . . ,Pn be properties of graphs. A (P1, . . . ,Pn)-partition of a graph G is a partition of the vertex set V (G) into subsets V1, . . . , Vn such that the subgraph G[Vi] induced by Vi has property Pi; i = 1, . . . , n. A graph G is said to be uniquely (P1, . . . ,Pn)-partitionable if G has exactly one (P1, . . . ,Pn)-partition. A property P is called hereditary if every subgraph of ev...

3. A lower bound To obtain a lower bound we count only those sequences of refinements which include the partition 1 .2.3 . . . d . r of n into d or d + 1 parts, d of which are of different size, where 0 < r = n-$d(d + 1) < d, so that J(2n) > d > J(2n)-3. Moreover we only count sequences in which we split off 1 from each of the d 1 parts of different size greater than 1. These d 1 steps can be m...