Asymptotic behaviour of the partition function

Abstract

Given a pair of positive integers and such that , for integer the quantity , called the partition function is considered; this by definition is equal to the cardinality of the set The properties of and its asymptotic behaviour as are studied. A geometric approach to this problem is put forward. It is shown that for sufficiently large , where and are positive constants depending on and , and and are characteristics of the exponential growth of the partition function. For some pair the exponents and are calculated as the logarithms of certain algebraic numbers; for other pairs the problem is reduced to finding the joint spectral radius of a suitable collection of finite-dimensional linear operators. Estimates of the growth exponents and the constants and are obtained.