On m–ary Partitions and Non-Squashing Stacks of Boxes

On m–ary Partitions and Non-Squashing Stacks of Boxes

The focus of this talk will be on congruences (divisibility properties) satisfied by various integer partition functions. I will share some history, starting with Ramanujan's groundbreaking work in the 1910's on the unrestricted partition function p(n) and moving rapidly to work by Robert Churchhouse in the late 1960's on the binary partition function. I will also discuss work of Oystein Rodseth, George Andrews, and Hansraj Gupta in the 1970's on results for m-ary partitions which are natural generalizations of binary partitions. (An m-ary partition of a positive integer n is a nonincreasing sequence of powers of m which sum to n. So, for example, 9+9+3+1+1+1 is a 3-ary partition of the integer 24.) I will then discuss recent work I completed with Rodseth which generalizes the results of Andrews and Gupta from the 1970's. I will close with a set of "applications" of m-ary partitions to Neil Sloane's questions on non-squashing stacks of boxes. Throughout the talk, I will attempt to discuss various aspects of the research related to symbolic computation. The talk will be self-contained and geared for a general audience.