Asymptotic behaviour of the partition function

Given a pair of positive integers m and d such that 2≤m≤d, for integer n≥0 the quantity bm,d(n), called the partition function is considered; this by definition is equal to the cardinality of the set. The properties of bm,d(n) and its asymptotic behaviour as n→∞ are studied. A geometric approach to this problem is put forward. It is shown that C1nλ1≤bm,d(n)≤C2nλ2, for sufficiently large n, where C1 and C2 are positive constants depending on m and d. For some pair (m,d) the exponents λ1 and λ2 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 C1 and C2 are obtained