On the size of the Durfee square of a random integer partition
For a partition \(\lambda = (\lambda_1,\lambda_2,\ldots)\), \(\lambda_1\geq \lambda_2\geq \cdots\), of a natural number \(n = \lambda_1+\lambda_2+\cdots\) let \(\eta(\lambda)\) denote the side length of the Durfee square of \(\lambda\). Given \(n\) and assuming that all \(p(n)\) partitions \(\lambda\) of \(n\) are equally likely, \(\eta_n = \eta(\cdot)\) may be considered as a random variable. The main result of this paper is to prove a local limit theorem for \(\eta_n\) of the form \[ P\left( \left( \eta_n - {\sqrt 6 \over \pi}\log 2 \sqrt n \right) \Biggl/n^{1/4} = v \right) = {n^{-1/4} \over \sqrt{2\pi}\sigma } e^{-v^2/(2\sigma^2)} (1+o(1)) \] as \(n\to\infty\) and uniformly for \(v = o(n^{1/12})\), where \[ \sigma^2 = {3\over 2}\sqrt 6 \left( {\pi^2 \over 6} - 2\log^2 2 \right)\Biggl/ \pi^3. \] The proof relies on the explicit generating function \[ 1 + \sum_{n=1}^\infty p(n) P(\eta_n = m) x^n = 2 x^{m^2} \prod_{j=1}^m {1\over (1-x^j)^2 } \] and on Cauchy's formula using the contour \[ |z|= 1 - {\pi \over \sqrt{6 n}} - {1\over n} + O(n^{-3/2}) \] and saddle point like approximation techniques. It should be noted---and this is also mentioned by the author---that \textit{E. R. Canfield, S. Cortell} and \textit{C. D. Savage} [Electron. J. Comb. 5, Research paper R32 (1998); printed version J. Comb. 5, 481-501 (1998; Zbl 0896.05007)] have obtained an asymptotic formula for \(P(\eta_n = y \sqrt n)\) for \(\varepsilon \leq y\leq 1-\varepsilon\) with a different method.
scientific article; zbMATH DE number 1787604