Fishburn diagrams, Fishburn numbers and their refined generating functions

The Fishburn numbers can be defined as the coefficients of the generating function 1 + ∑ m = 1 ∞ ∏ i = 1 m ( 1 − ( 1 − t ) i ) . Combinatorially, the Fishburn numbers enumerate certain supersets of sets enumerated by the Catalan numbers. We add to this work by giving an involution-based proof of the conjecture of Claesson and Linusson that the Fishburn numbers enumerate non-2-neighbor-nesting matchings. We begin by proving that a map originally defined by Claesson and Linusson gives a bijection between non-2-neighbor-nesting matchings and ( 2 − 1 ) -avoiding inversion tables. We then define a set of diagrams, which we call Fishburn diagrams, that give a natural interpretation to the generating function of the Fishburn numbers. Using an involution on Fishburn diagrams, we then prove that the Fishburn numbers enumerate ( 2 − 1 ) -avoiding inversion tables. By using two variations of this involution on two subsets of Fishburn diagrams, we then give a visual proof of the conjecture of Remmel and Kitaev that two bivariate refinements of the generating function of the Fishburn numbers are equivalent. In Appendix A, we give an inductive proof of the conjecture of Claesson and Linusson that the distribution of left-nestings over the set of all matchings is given by the second-order Eulerian triangle. njecture of Remmel and Kitaev was independently proved by Jelinék and by Yan with a matrix interpretation defined by Dukes and Parviainen. Bijections surveyed by Callan can lead to a similar proof of the conjecture of Claesson and Linusson giving the distribution of left-nestings over matchings, using a result on the Stirling permutations due to Gessel and Stanley. This work was done independently.