On Fishburn numbers

Vít Jelínek, 17 Apr 2023

Fishburn numbers (sequence A022493 in the OEIS) are a remarkable counting sequence with many different combinatorial interpretations. I will give an overview of several combinatorial structures enumerated by these numbers, which include posets with no ‘2+2’ subposet, upper-triangular matrices with no zero rows or columns, permutations avoiding a certain pattern, as well as certain families of finite integer sequences. I will also talk about combinatorial and algebraic properties of the Fishburn numbers, with particular focus on their relationship with the much better known Catalan numbers.