Statistical theory of integer partitions

Paper presented at Strathmore International Math Research Conference on July 23 - 27, 201

On the number of summands in a random prime partition

Paper presented at Strathmore International Math Research Conference on July 23 - 27, 2012We study the length (number of summands) in partitions of an integer into primes, both in the restricted (unequal summands) and unrestricted case. It is shown how one can obtain asymptotic expansions for the mean and variance (and potentially higher moments), which is in contrast to the fact that there is no asymptotic formula for the number of such partitions in terms of elementary functions. Building on ideas of Hwang, we also prove a central limit theorem in the restricted case. The technique also generalizes to partitions into powers of primes, or even more generally, the values of a polynomial at the prime numbers.We study the length (number of summands) in partitions of an integer into primes, both in the restricted (unequal summands) and unrestricted case. It is shown how one can obtain asymptotic expansions for the mean and variance (and potentially higher moments), which is in contrast to the fact that there is no asymptotic formula for the number of such partitions in terms of elementary functions. Building on ideas of Hwang, we also prove a central limit theorem in the restricted case. The technique also generalizes to partitions into powers of primes, or even more generally, the values of a polynomial at the prime numbers

On the distribution of multiplicities in integer partitions

Paper presented at Strathmore International Math Research Conference on July 23 - 27, 2012We study the distribution of the number of parts of given multiplicity (or equivalently ascents of given size) in integer partitions. In this paper we give methods to compute asymptotic formulas for the expected value and variance of the number of parts of multiplicity d (d is a positive integer) in a random partition of a large integer n and also prove that the limiting distribution is asymptotically normal for fixed d. However, if we let d increase with n, we get a phase transition for d around n1=4. Our methods can also be applied to so called -partitions where the parts are members of a sequence of integers

Asymptotic Normality of Almost Local Functionals in Conditioned Galton-Watson Trees

An additive functional of a rooted tree is a functional that can be calculated recursively as the sum of the values of the functional over the branches, plus a certain toll function. Janson recently proved a central limit theorem for additive functionals of conditioned Galton-Watson trees under the assumption that the toll function is local, i.e. only depends on a fixed neighbourhood of the root. We extend his result to functionals that are almost local, thus covering a wider range of functionals. Our main result is illustrated by two explicit examples: the (logarithm of) the number of matchings, and a functional stemming from a tree reduction process that was studied by Hackl, Heuberger, Kropf, and Prodinger

Counting Planar Tanglegrams

Tanglegrams are structures consisting of two binary rooted trees with the same number of leaves and a perfect matching between the leaves of the two trees. We say that a tanglegram is planar if it can be drawn in the plane without crossings. Using a blend of combinatorial and analytic techniques, we determine an asymptotic formula for the number of planar tanglegrams with n leaves on each side