The Burge correspondence and crystal graphs

The Burge correspondence yields a bijection between simple labelled graphs and semistandard Young tableaux of threshold shape. We characterize the simple graphs of hook shape by peak and valley conditions on Burge arrays. This is the first step towards an analogue of Schensted's result for the RSK insertion which states that the length of the longest increasing subword of a word is the length of the largest row of the tableau under the RSK correspondence. Furthermore, we give a crystal structure on simple graphs of hook shape. The extremal vectors in this crystal are precisely the simple graphs whose degree sequence are threshold and hook-shaped.Comment: 19 pages; final version to appear in European Journal of Combinatoric

Uncrowding algorithm for hook-valued tableaux

Whereas set-valued tableaux are the combinatorial objects associated to stable Grothendieck polynomials, hook-valued tableaux are associated to stable canonical Grothendieck polynomials. In this paper, we define a novel uncrowding algorithm for hook-valued tableaux. The algorithm ``uncrowds'' the entries in the arm of the hooks and yields a set-valued tableau and a column-flagged increasing tableau. We prove that our uncrowding algorithm intertwines with crystal operators. An alternative uncrowding algorithm that ``uncrowds'' the entries in the leg instead of the arm of the hooks is also given. As an application of uncrowding, we obtain various expansions of the canonical Grothendieck polynomials.Comment: 32 page

Promotion and growth diagrams for fans of Dyck paths and vacillating tableaux

We construct an injection from the set of $r$-fans of Dyck paths (resp. vacillation tableaux) of length $n$ into the set of chord diagrams on $[n]$ that intertwines promotion and rotation. This is done in two different ways, namely as fillings of promotion--evacuation diagrams and in terms of Fomin growth diagrams. Our analysis uses the fact that $r$-fans of Dyck paths and vacillating tableaux can be viewed as highest weight elements of weight zero in crystals of type $B_r$ and $C_r$, respectively, which in turn can be analyzed using virtual crystals. On the level of Fomin growth diagrams, the virtualization process corresponds to Krattenthaler's blow up construction. One of the motivations for finding rotation invariant diagrammatic bases such as chord diagrams is the cyclic sieving phenomenon. Indeed, we give a cyclic sieving phenomenon on $r$-fans of Dyck paths and vacillating tableaux using the promotion action.Comment: 40 pages, 13 figure

Catalan and Crystal Combinatorics

The Catalan numbers are a ubiquitous sequence of natural numbers appearing in a diversearray of mathematical fields. However, even though these numbers have been well-studied, several conjectures and properties surrounding the Catalan numbers remain open. In this dissertation we first study the joint distribution of various statistics defined on Dyck paths. The first joint distribution involves the area and diagonal inversion statistic in the form of the q, t-Catalan polynomial. This polynomial arises from the study of the space of diagonal harmonics, and its symmetry has evaded a combinatorial proof. We introduce two new q, t-Catalan polynomials using two new statistics on Dyck paths. We are able to give a combinatorial proof of their symmetry and recover the usual q, t-Catalan polynomial in terms of our new statistics. Next, we explore the joint distribution of NE and NNE-factors within Dyck paths. We answer an open question by Bóna and Labelle regarding the symmetry of these numbers at certain values. Additionally, we prove various enumerative results of these numbers, including their real-rootedness and their connection to the number of cyclic compositions.Kashiwara’s crystal bases are combinatorial structures introduced in his study of the representations of quantum groups under a certain limit. Using Kashiwara’s crystals, we explore the Burgecorrespondence sending labelled graphs to tableau. We give a Schensted-like result characterizing when a labelled graph is sent to a hook-shaped tableau and give a type A crystal structure on such graphs. Lastly, we merge these two topics by looking at the space of invariant tensors of the spin and vector representations in Type B. Using the promotion operator on Kashiwara’s crystals, we construct a diagrammatic basis for these spaces in terms of chord diagrams such that rotation of the chord diagrams intertwines with the cyclic action on tensor factors. As a consequence of this, we are able to give a cyclic sieving phenomenon for fans of Dyck paths and vacillating tableaux respectively

An Area-Depth Symmetric q,tq,t-Catalan Polynomial

We define two symmetric $q,t$-Catalan polynomials in terms of the area and depth statistic and in terms of the dinv and dinv of depth statistics. We prove symmetry using an involution on plane trees. The same involution proves symmetry of the Tutte polynomials. We also provide a combinatorial proof of a remark by Garsia et al. regarding parking functions and the number of connected graphs on a fixed number of vertices

Uncrowding algorithm for hook-valued tableaux

Whereas set-valued tableaux are the combinatorial objects associated to stable Grothendieck polynomials, hook-valued tableaux are associated to stable canonical Grothendieck polynomials. In this paper, we define a novel uncrowding algorithm for hook-valued tableaux. The algorithm "uncrowds" the entries in the arm of the hooks and yields a set-valued tableau and a column-flagged increasing tableau. We prove that our uncrowding algorithm intertwines with crystal operators. An alternative uncrowding algorithm that "uncrowds" the entries in the leg instead of the arm of the hooks is also given. As an application of uncrowding, we obtain various expansions of the canonical Grothendieck polynomials