On sum-and-distance systems, reversible square matrices and divisor functions

The event is taking part on the Tuesday, Oct 23rd 2018 at 15:30
Theme/s: Pure Maths
Location of Event: Alan Turing Building Room 306
This event is a: Public Seminar

Abstract: A d-dimensional sum-and-distance system is a collection of d finite sets of integers such that the sums and differences formed by taking one element from each set generate a prescribed arithmetic progression.

In the talk, I will discuss how for a fixed dimension n, there exists a bijection between the set of principal reversible square matrices and the set of 2-dimensional sum-and-distance systems. The number of 2-dimensional sum-and-distance systems can then be obtained by using Ollerenshaw and Bree's enumeration argument for the number of principal reversible square matrices. By reworking their argument one finds that the number of such principal reversible square matrices, (and so 2-dimensional sum-and-distance systems) can be expressed in terms of sums of products of the j-th non-trivial divisor function c_j(n).; this counts the total number of proper ordered factorisations of the integer n into j factors (i.e. excluding the factor 1, but counting permutations of the factors). If j is greater than the total number of prime factors of n, including repeats, then c_j(n) = 0. The non-trivial divisor function c_j(n) has been far less studied than its multiplicative cousin, the j-th divisor function d_j(n) .
Further relations concerning these two functions are discussed in the talk.

This is joint work with Sally Hill, Martin Huxley and Karl Michael Schmidt.

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External Speakers

Matthew Lettington (University of Cardiff)