@article{Benoumhani2025, 
author = {Moussa Benoumhani},
title = {Restricted partitions and convex topologies},
year = {2025},
journal = {AIMS Mathematics},
volume = {10},
number = {4},
pages = {10187-10203},
keywords = {central limit theorem, Fibonacci number, Lucas number, partition, polynomial, unimodal},
url = {https://www.sciopen.com/article/10.3934/math.2025464},
doi = {10.3934/math.2025464},
abstract = {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          o      c        (  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.}
}