Given a pair of positive integers m and d such that 2 â¤ m â¤ d, for integer n â¥ 0 the quantity bm,d(n), called the partition function is considered; this by definition is equal to the cardinality of the set (a0, a1, . . .) : n = Î£kakmk, ak â 0, . . . , d - 1, k â¥ 0. The properties of bm,d(n) and its asymptotic behaviour as n â â are studied. A geometric approach to this problem is put forward. It is shown that C1nÎ»1 â¤ bm,d(n) â¤ C2nÎ»2 for sufficiently large n, where C1 and C2 are positive constants depending on m and d, and Î»1 = lim/nââ log b(n)/log n and Î»2 = limnââ log b(n)/log n are characteristics of the exponential growth of the partition function. For some pair (m, d) the exponents Î»1 and Î»2 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.
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