Some central limit theorems for doubly restricted partitions

<p>Let P_{K, L}(N) be the number of <u>unordered</u> partitions of a positive integer N into K or fewer positive integer parts, each part not exceeding L. A distribution of the form</p> <p>Ʃ/N≤x P_K,L(N)</p> <p>is considered first. For any fixed K, this distribution approaches a piecewise polynomial function as L increases to infinity. As both K and L approach infinity, this distribution is asymptotically normal. These results are proved by studying the convergence of the characteristic function.</p> <p>The main result is the asymptotic behavior of P_K,K(N) itself, for certain large K and N. This is obtained by studying a contour integral of the generating function taken along the unit circle. The bulk of the estimate comes from integrating along a small arc near the point 1. Diophantine approximation is used to show that the integral along the rest of the circle is much smaller. </p>