Asymptotics of the Number Partitioning Distribution

The number partitioning problem can be interpreted physically in terms of a thermally isolated non-interacting Bose gas trapped in a one-dimensional harmonic oscillator potential. We exploit this analogy to characterize, by means of a detour to the Bose gas within the canonical ensemble, the probability distribution for finding a specified number of summands in a randomly chosen partition of an integer n. It is shown that this distribution approaches its asymptotics only for n > 10. Consider the decompositions of a natural number n into natural summands, without regard to order. Let Φ(n,M) denote the number of such partitions which consist of M parts, and Ω(n) = ∑n M=1 Φ(n,M) the total number of partitions. For n = 4, for instance, we have 4 = 1 + 1 + 1 + 1 = 2 + 1 + 1 = 2 + 2 = 3 + 1 , (1) hence Φ(4, 4) = 1, Φ(4, 3) = 1, Φ(4, 2) = 2, Φ(4, 1) = 1, adding up to Ω(4) = 5. It is known that Ω(n) grows exponentially with √ n [1], so that the enumeration of the individual partitions soon becomes impractical when n gets larger. It is then useful to focus on the distribution pmc(n,M) ≡ Φ(n,M) Ω(n) (0 ≤ M ≤ n) , (2) which gives the probability for finding M summands in a randomly chosen partition of n. For moderately large n, this distribution can be computed numerically with the help of the recursion relation Φ(n,M) = min{n−M,M}