Restricted partitions and convex topologies

Let $X_n$ be a finite set. We consider two types of sequences of restricted partitions of $X_n$, namely, the number of order consecutive partitions of $X_n$ into $k$ parts, denoted $N_{oc}(n,k)$ and the sequence $T(n,k)$ of the number of order-consecutive partition sequences of $X_n$ with $k$ parts. This last sequence is also the number of locally convex topologies consisting of $k$ nested open sets defined on a totally ordered set of cardinality $n$. Although all the main results apply to both sequences, we will focus on $T(n,k)$. We prove that the generating polynomials of these sequences have real negative roots. A central limit theorem and a local limit theorem are also proved for $T(n,k)$. Many other relations with Fibonacci and Lucas numbers are also given.