Martin Rubey

Ptolemy diagrams and torsion pairs in the cluster categories of Dynkin type D

We give a complete classification of torsion pairs in the cluster category of Dynkin type D_n, via a bijection to new combinatorial objects called Ptolemy diagrams of type D. For the latter we give along the way different combinatorial descriptions. One of these allows us to count the number of torsion pairs in the cluster category of type D_n by providing their generating function explicitly.

Torsion pairs in cluster tubes

We give a complete classification of torsion pairs in the cluster categories associated to tubes of finite rank. The classification is in terms of combinatorial objects called Ptolemy diagrams which already appeared in our earlier work on torsion pairs in cluster categories of Dynkin type A. As a consequence of our classification we establish closed formulae enumerating the torsion pairs in cluster tubes, and find that the torsion pairs in cluster tubes exhibit a cyclic sieving phenomenon.

Crossings, Motzkin paths and moments

Kasraoui, Stanton and Zeng, and Kim, Stanton and Zeng introduced certain q-analogues of Laguerre and Charlier polynomials. The moments of these orthogonal polynomials have combinatorial models in terms of crossings in permutations and set partitions. The aim of this article is to prove simple formulae for the moments of the q-Laguerre and the q-Charlier polynomials, in the style of the Touchard-Riordan formula (which gives the moments of some q-Hermite polynomials, and also the distribution of crossings in matchings). Our method mainly consists of the enumeration of weighted Motzkin paths, which are naturally associated with the moments. Some steps are bijective, in particular, we describe a decomposition of paths which generalises a previous construction of Penaud for the case of the Touchard-Riordan formula. There are also some non-bijective steps using basic hypergeometric series, and continued fractions or, alternatively, functional equations.

A Determinantal Formula for the Hilbert Series of One-sided Ladder Determinantal Rings

We give a formula that expresses the Hilbert series of one-sided ladder determinantal rings, up to a trivial factor, in form of a determinant. This allows the convenient computation of these Hilbert series. The formula follows from a determinantal formula for a generating function for families of nonintersecting lattice paths that stay inside a one-sided ladder-shaped region, in which the paths are counted with respect to turns.

Symmetries of statistics on lattice paths between two boundaries

Abstract We prove that on the set of lattice paths with steps N = ( 0 , 1 ) and E = ( 1 , 0 ) that lie between two fixed boundaries T and B (which are themselves lattice paths), the statistics ‘number of E steps shared with B’ and ‘number of E steps shared with T’ have a symmetric joint distribution. To do so, we give an involution that switches these statistics, preserves additional parameters, and generalizes to paths that contain steps S = ( 0 , − 1 ) at prescribed x-coordinates. We also show that a similar equidistribution result for path statistics follows from the fact that the Tutte polynomial of a matroid is independent of the order of its ground set. We extend the two theorems to k-tuples of paths between two boundaries, and we give some applications to Dyck paths, generalizing a result of Deutsch, to watermelon configurations, to pattern-avoiding permutations, and to the generalized Tamari lattice. Finally, we prove a conjecture of Nicolas about the distribution of degrees of k consecutive vertices in k-triangulations of a convex n-gon. To achieve this goal, we provide a new statistic-preserving bijection between certain k-tuples of non-crossing paths and k-flagged semistandard Young tableaux, which is based on local moves reminiscent of jeu de taquin.

Combinatorial interpretations of the q -Faulhaber and q -Salié coefficients

Recently, Guo and Zeng discovered two families of polynomials featuring in a q-analogue of Faulhabers formula for the sums of powers and a q-analogue of Gessel-Viennots formula involving Salies coefficients for the alternating sums of powers. In this paper, we show that these are polynomials with symmetric, nonnegative integral coefficients by refining Gessel-Viennots combinatorial interpretations.