A CONVERGENCE CRITERION FOR RECURRENT SEQUENCES WITH APPLICATION TO THE PARTITION LATTICE
László Babai, Tamás Lengyel · Analysis · 1992
. We prove a fairly general convergence criterion for sequences satisfying a linear recurrence (defined by an infinite triangular matrix). We prove that every sequence of positive numbers satisfying a nearly convex linear recurrence with finite retardation and active predecessors converges to a positive limit. -- Informally, near convexity means the coefficients are nonnegative and the sum of coefficients in each equation is approximately 1; finite retardation means low order terms have little weight; and active predecessors mean that the immediate predecessor carries a weight greater than a fixed positive constant. -- We present an application to the asymptotic number of not necessarily maximal chains in the partition lattice. The coefficients of the corresponding recurrence are the Stirling numbers of the second kind. AMS 1980 classification numbers: 40A05, 11B37, 05A15, 06C10, 11B73 1. Introduction The use of recurrence relations is one of the classical methods in combinatorial enu...