The r-complete partitions

Minimal r-Complete Partitions

A minimal r-complete partition of an integer m is a partition of m with as few parts as possible, such that all the numbers 1, . . . , rm can be written as a sum of parts taken from the partition, each part being used at most r times. This is a generalization of M-partitions (minimal 1-complete partitions). The number of M-partitions of m was recently connected to the binary partition function ...

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.

The Double-complete Partitions of Integers

Representing a positive integer in terms of a sum of smaller numbers with certain conditions has been studied since MacMahon [5] pioneered perfect partitions. The complete partitions is in this category and studied by the second author [6]. In this paper, we study complete partitions with more specified completeness, which we call the double-complete partitions.

T-partitions and S-complete T-partitions of a Graph

Cyclic partitions of complete nonuniform hypergraphs and complete multipartite hypergraphs

A cyclic q-partition of a hypergraph (V,E) is a partition of the edge set E of the form {F, F , F θ 2 , . . . , F θ q−1 } for some permutation θ of the vertex set V . Let Vn = {1, 2, . . . , n}. For a positive integer k, ( Vn k ) denotes the set of all k-subsets of Vn. For a nonempty subset K of Vn−1, we let K n denote the hypergraph ( Vn, ⋃ k∈K ( Vn k )) . In this paper, we find a necessary an...