On the size of the Durfee square of a random integer partition

We prove a local limit theorem for the length of the side of the Durfee square in a random partition of a positive integer n as n→∞. We rely our asymptotic analysis on the power series expansion of xm2∏j=1m(1−xj)−2 whose coefficient of xn equals the number of partitions of n in which the Durfee square is m2.