Multiplicative Partition Functions Sample Clauses

Multiplicative Partition Functions. ‌ A partition of a natural number n is a finite weakly decreasing sequence of positive integers that sums to n. For k ∈ N, k > 1, let pk(n) count the k-regular partitions of n, i.e., partitions of n for which no part is divisible by k. These generating functions arise in many different contexts, in particular in connection with the representation theory of the symmetric groups, Hecke algebras, and related groups and algebras; for a long time, this has been studied both in combinatorics and number theory. For the classical (unrestricted) partition function p(n), explicit formulae are known due to the work of Hardy, Ramanujan and Xxxxxxxxxx, and more recent work of Bruinier and Ono [BO13]. Based on a result due to Xxxxxx, the following inequality was shown in a recent article by the Bessendrodt and Ono [BO16]: For any integers a, b such that a, b > 1 and a + b > 9, we have p(a)p(b) > p(a + b). Y Also the cases of equality were determined in [BO16]. The inequality above was then used to study an “extended partition function”, given by defining for a partition µ = (µ1, µ2, . . .): p(µ) = p(µj). With P (n) denoting the set of all partitions of n, the maximum maxp(n) = max(p(µ) | µ ∈ P (n)) was determined explicitly in [BO16]; see Chapter 5 for the complete statement. Y In Chapter 5, I present the results of a joint project with Xxxxxxxxx Xxxxxxxxxx, in which we obtain results analogous to the theorems [BO16] for pk(n). My contribution to this project was proving a result corresponding to the inequality above for an extension of the generating function pk(n) to a function on the set Pk(n) of all k- regular partitions of n, defined for µ = (µ1, µ2, . . .) ∈ Pk(n) by: pk(µ) = pk(µj).