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arXiv · 2407.04409

The Fibonacci-Fubini and Lucas-Fubini numbers

Yahia Djemmada·Abdelghani Mehdaoui·László Németh·László Szalay
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Abstract

Based on the combinatorial interpretation of the ordered Bell numbers, which count all the ordered partitions of the set $[n]=\{1,2,\dots,n\}$, we introduce the Fibonacci partition as a Fibonacci permutation of its blocks. Then we define the Fibonacci-Fubini numbers that count the total number of Fibonacci partitions of $[n]$. We study the classical properties of this sequence (generating function, explicit and Dobi\'nski-like formula, etc.), we give combinatorial interpretation, and we extensively examine the Fibonacci-Fubini arithmetic triangle. We give some associate linear recurrence sequences, where in some sequences the Stirling numbers of the first and second kinds appear as well.

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Yahia Djemmada, Abdelghani Mehdaoui, László Németh, László Szalay. 2024-07-05. The Fibonacci-Fubini and Lucas-Fubini numbers. https://arxiv.org/abs/2407.04409

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