The Fibonacci-Fubini and Lucas-Fubini numbers

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.