On naturally labelled posets and permutations avoiding 12-34

A partial order $\prec$ on $[n]$ is naturally labelled (NL) if $x\prec y$ implies $x<y$. We establish a bijection between {3, 2+2}-free NL posets and 12-34-avoiding permutations, determine functional equations satisfied by their generating function, and use series analysis to investigate their asymptotic growth. We also exhibit bijections between 3-free NL posets and various other objects, and determine their generating function. The connection between our results and a hierarchy of combinatorial objects related to interval orders is described.Comment: 19 page

Refining the bijections among ascent sequences, (2+2)-free posets, integer matrices and pattern-avoiding permutations

The combined work of Bousquet-M\'elou, Claesson, Dukes, Jel\'inek, Kitaev, Kubitzke and Parviainen has resulted in non-trivial bijections among ascent sequences, (2+2)-free posets, upper-triangular integer matrices, and pattern-avoiding permutations. To probe the finer behavior of these bijections, we study two types of restrictions on ascent sequences. These restrictions are motivated by our results that their images under the bijections are natural and combinatorially significant. In addition, for one restriction, we are able to determine the effect of poset duality on the corresponding ascent sequences, matrices and permutations, thereby answering a question of the first author and Parviainen in this case. The second restriction should appeal to Catalaniacs.Comment: 24 pages, 4 figures. To appear in Journal of Combinatorial Theory, Series

Combinatorial Structures and Generating Functions of Fishburn Numbers, Parking Functions, and Tesler Matrices.

This dissertation reflects the author\u27s work on two problems involving combinatorial structures. The first section, which was also published in the Journal of Combinatorial Theory, Series A, discusses the author\u27s work on several conjectures relating to the Fishburn numbers. The Fishburn numbers can be defined as the coefficients of the generating function begin{align*} 1+sum_{m=1}^{infty} prod_{i=1}^{m}(1-(1-t)^{i}). end{align*} 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 $textbf{(2-1)}$-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 $textbf{(2-1)}$-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 an appendix, 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. The conjecture of Remmel and Kitaev was independently proved by Jelin\u27ek 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. The second section, some of which was presented at the Formal Power Series and Algebriac Combinatorics conference (FPSAC), discusses the author\u27s work on several conjectures relating to parking functions and to Tesler matrices. Parking functions are combinatorial objects which generalize both permutations and Catalan paths. Haglund and Loehr conjectured that the generating functions of two statistics, $area$ and $dinv$, over the set of parking functions (the $q, t$-parking functions) gives the Hilbert series of the diagonal coinvariants. Haglund recently proved that this Hilbert series is given by another generating function over the set of matrices with every hook sum equal to one (``Tesler matrices\u27\u27). We prove several structural results on parking functions inspired by Tesler matrices, including a near-recursive generation of the $q, t$-parking functions. We also give consistent bijective proofs of several special cases of Haglund\u27s Tesler function identity, giving a combinatorial connection between parking functions and Tesler matrices, and discuss related conjectures. A connection between the $q=1$ special case and a result of Kreweras was later pointed out by Garsia et al, and some of the original ideas on the $q=1$ special case arose from a discussion between the author, Haglund, Bandlow, and Visontai