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.
Citation: Moussa Benoumhani. Restricted partitions and convex topologies[J]. AIMS Mathematics, 2025, 10(4): 10187-10203. doi: 10.3934/math.2025464
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.
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Moussa Benoumhani. Restricted partitions and convex topologies[J]. AIMS Mathematics, 2025, 10(4): 10187-10203. doi: 10.3934/math.2025464
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