Bounds on the number of generalized partitions and some applications

We present bounds concerning the number of Hartmanis partitions of a finite set. An application of these inequalities improves the known asymptotic lower bound on the number of linear spaces on n points. We also present an upper bound for a certain class of these partitions which bounds the number of Steiner triple and quadruple systems. Recent work has extended the known numerical values for the number of linear spaces on n points [1]. Upper and lower bounds of 2 (n 3) and 2 n , respectively, were given in [2, 7]. We improve the lower bound by showing an inequality of Knuth (see [5]) to hold for more general structures known as Hartmanis d-partitions. We prove an upper bound for the number of these structures and also give an upper bound for a certain class of these partitions. This last inequality gives asymptotic upper bounds for the number of Steiner triple and quadruple systems. A linear space is a collection of points and lines such that every pair of distinct points are on a unique line and every line contains at least two points. Let S n := {1,. .. , n} be a finite set of size n and S d n the collection of all d-element subsets of S n. We say that a collection of subsets P of S n is a d-partition of S n (to be more specific, a Hartmanis d-partition, see [4]) if