Tutte seminar - Bruce Richmond

Maximum Stirling Numbers of the Second Kind

Abstract:

Say an integer n is exceptional if the maximum Stirling number of the second kind S(n,k) occurs for two (of necessity consecutive) values of k. We prove that the number of exceptional integers less than or equal to x is O(x^ε), for any ε > 0. A theorem of Bombieri and Pila estimating lattice points on a smooth curve is essential. An estimate for the number of intersections of an analytic curve with a polynomial of degree d is also essential. The theory offewnomials as developed by Khovanskii seems essential. The exact result we need is found in a paper of Gwozdziewicz, K. Kurdyka and A. Parusinski estimating the number of solutions of an algebraic equation on the curve y = e^x + sin x, x > 0. J. Pila provided this reference. 

This is joint work with R. Canfield, G. Kemkes, D. Merlini and C. Pomerance.