Combinatorial aspects of symmetries on groups

An MSc dissertation by Shivani Singh. University of Witwatersrand Faculty of Science, School of Mathematics. August 2016.These symmetries have interesting applications to enumerative combinatorics and to Ramsey theory. The aim of this thesis will be to present some important results in these fields. In particular, we shall enumerate the r-ary symmetric bracelets of length n.LG201

Modular Schur numbers

For any positive integers l and m, a set of integers is said to be (weakly) l-sumfree modulomif it contains no (pairwise distinct) elements x1,x2,...,xl,y satisfying the congruence x1+...+xl ≡ y mod m. It is proved that, for any positive integers k and l, there exists a largest integer n for which the set of the first n positive integers {1,2,...,n} admits a partition into k (weakly) l-sum-free sets modulo m. This number is called the generalized (weak) Schur number modulo m, associated with k and l. In this paper, for all positive integers k and l, the exact value of these modular Schur numbers are determined for m = 1, 2 and 3