Matthew Lettington (combinatorics)

17-01-2020 at 2pm to 3pm

Location: J325

Audience: Public

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Title: Divisor Functions and Sum Systems


Abstract: Divisor functions have attracted the attention of number theorists from Dirichlet to the present day. Here we consider associated divisor functions c_j^{(r)}(n) which for non-negative integers j,r count the number of ways of representing n as an ordered product of j+r factors, of which the first j must be non-trivial, and their natural extension to negative integers r. We give recurrence properties and explicit formulae for these novel arithmetic functions. Specifically, the functions c_j^{(-j)}(n) count, up to a sign, the number of ordered factorisations of n into j square-free non-trivial factors. These functions are related to a modified version of the Mobius function and turn out to play a central role in counting the number of sum systems of given dimensions.

 

Sum systems are finite collections of finite sets of non-negative integers, of prescribed cardinalities, such that their set sum generates consecutive integers without repetitions. Using a recently established bijection between sum systems and joint ordered factorisations of their component set cardinalities, we prove a formula expressing the number of different sum systems in terms of associated divisor functions.