Related papers: Major Index on Catalan Combinatorics

We introduce a notion of Dyck paths with coloured ascents. For several ways of colouring, we establish bijections between sets of such paths and other combinatorial structures, such as non-crossing trees, dissections of a convex polygon,…

We study the convergence of distributions on finite paths of weighted digraphs, namely the family of Boltzmann distributions and the sequence of uniform distributions. Targeting applications to the convergence of distributions on paths, we…

We investigate nil-Temperley-Lieb algebras of type A. We give a general description of the structure of monomials formed by the generators. We also show that the dimensions of these algebras are the famous Catalan numbers by providing a…

Instead of $k$-Dyck paths we consider the equivalent concept of $k$-non-crossing trees. This is our preferred approach relative to down-step statistics modulo $k$ (first studied by Heuberger, Selkirk, and Wagner by different methods). One…

The $q,t$-Catalan numbers can be defined using rational functions, geometry related to Hilbert schemes, symmetric functions, representation theory, Dyck paths, partition statistics, or Dyck words. After decades of intensive study, it was…

We study the Fuss--Catalan algebras, which are generalizations of the Temperley--Lieb algebra and act on generalized Dyck paths, through non-crossing partitions. First, the Temperley--Lieb algebra is defined on non-crossing partitions, and…

We prove that the pair of statistics (des,maj) on multiset permutations is equidistributed with the pair (stc,inv) on certain quotients of the symmetric group. We define the analogue of the statistic stc on multiset permutations, whose…

In this paper we develop a systematic calculus for the Duistermaat index, a symplectic invariant defined for triples of Lagrangian subspaces. Introduced nearly half a century ago, this index has lately been the subject of renewed attention,…

We consider a family of infinite sums of products of Catalan numbers, indexed by trees. We show that these sums are polynomials in $1/\pi$ with rational coefficients; the proof is effective and provides an algorithm to explicitly compute…

We study the distribution of the statistics 'number of fixed points' and 'number of excedances' in permutations avoiding subsets of patterns of length 3. We solve all the cases of simultaneous avoidance of more than one pattern, giving…

In this paper we generalize the notion of the comparative index for the pair of Lagrangian subspaces which has fundamental applications in oscillation theory of symplectic difference systems and linear differential Hamiltonian systems. We…

Let $k \geq 2$ be an integer. We prove that factorization of integers into $k$ parts follows the Dirichlet distribution $\text{Dir}\left(\frac{1}{k},\ldots,\frac{1}{k}\right)$ by multidimensional contour integration, thereby generalizing…

We introduce new natural generalizations of the classical descent and inversion statistics for permutations, called width-$k$ descents and width-$k$ inversions. These variations induce generalizations of the excedance and major statistics,…

Path pairs are a modification of parallelogram polyominoes that provide yet another combinatorial interpretation of the Catalan numbers. More generally, the number of path pairs of length $n$ and distance $\delta$ corresponds to the…

We study certain types of ideals in the standard Borel subalgebra of an untwisted affine Lie algebra. We classify these ideals in terms of the root combinatorics and give an explicit formula for the number of such ideals in type $A$. The…

In the paper, with the aid of the series expansions of the square or cubic of the arcsine function, the authors establish several possibly new combinatorial identities containing the ratio of two central binomial coefficients which are…

In this paper we determine the parity of some sequences which are related to Catalan numbers. Also we introduce a combinatorical object called, \Catalan tree", and discuss its properties.

Given a permutation statistic $\operatorname{st}$, define its inverse statistic $\operatorname{ist}$ by $\operatorname{ist}(\pi):=\operatorname{st}(\pi^{-1})$. We give a general approach, based on the theory of symmetric functions, for…

In this paper we study the poset of basic tilting $kQ$-modules when $Q$ is a Dynkin quiver, and the poset of basic support $\tau$-tilting $kQ$-modules when $Q$ is a connected acyclic quiver respectively. It is shown that the first poset is…

The study of Mahonian statistics dated back to 1915 when MacMahon showed that the major index and the inverse number have the same distribution on a set of permutations with length n. Since then, many Mahonian statistics have been…