Counting Dyck paths by area and rank

The set of Dyck paths of length $2n$ inherits a lattice structure from a bijection with the set of noncrossing partitions with the usual partial order. In this paper, we study the joint distribution of two statistics for Dyck paths: \emph{area} (the area under the path) and \emph{rank} (the rank in the lattice). While area for Dyck paths has been studied, pairing it with this rank function seems new, and we get an interesting $(q,t)$-refinement of the Catalan numbers. We present two decompositions of the corresponding generating function: one refines an identity of Carlitz and Riordan; the other refines the notion of $\gamma$-nonnegativity, and is based on a decomposition of the lattice of noncrossing partitions due to Simion and Ullman. Further, Biane's correspondence and a result of Stump allow us to conclude that the joint distribution of area and rank for Dyck paths equals the joint distribution of length and reflection length for the permutations lying below the $n$-cycle $(12...n)$ in the absolute order on the symmetric group.Comment: 24 pages, 7 figures. Connections with work of C. Stump (arXiv:0808.2822v2) eliminated the need for 5 pages of proof in the first draf

From Dürer's Magic Square to Klumpenhouwer Tesseracts: On Melencolia (2013) by Philippe Manoury

Many Western art music composers have taken advantage of tabulated data for nourishing their creative practices, particularly since the early twentieth century. The arrival of atonality and serial techniques was crucial to this shift. Among the authors dealing with these kinds of tables, some have considered the singular mathematical properties of magic squares. This paper focuses on a particular case study in this sense: Philippe Manoury's Third String Quartet, entitled Melencolia. We mainly analyse mainly several strategies conceived by the French composer – through his own sketches – in order to manipulate pitches and pitch-classes over time. For that purpose, we take advantage of Klumpenhouwer networks as a way to settle wide and dense isographic relationships. Our hyper-K-nets sometimes reach a total of 32 arrows that allow geometrical arrangements as tesseracts in which their different dimensions cluster related families of isographies. In doing so, we aim to provide an instructive example of how to contextualise K-nets and isographies as powerful tools for the analysis of compositional practices

Geometric and Topological Combinatorics

The 2007 Oberwolfach meeting “Geometric and Topological Combinatorics” presented a great variety of investigations where topological and algebraic methods are brought into play to solve combinatorial and geometric problems, but also where geometric and combinatorial ideas are applied to topological questions

Some multivariate master polynomials for permutations, set partitions, and perfect matchings, and their continued fractions

We find Stieltjes-type and Jacobi-type continued fractions for some "master polynomials" that enumerate permutations, set partitions or perfect matchings with a large (sometimes infinite) number of simultaneous statistics. Our results contain many previously obtained identities as special cases, providing a common refinement of all of them.Comment: LaTeX2e, 122 pages, includes 9 tikz figure