相关论文: Dinv and Area

In this paper we consider the probabilistic theory of Diophantine approximation in projective space over a completion of Q. Using the projective metric studied by Bombieri, van der Poorten, and Vaaler we prove the analogue of Khintchine's…

A classical result of Dowker (Bull. Amer. Math. Soc. 50: 120-122, 1944) states that for any plane convex body $K$, the areas of the maximum (resp. minimum) area convex $n$-gons inscribed (resp. circumscribed) in $K$ is a concave (resp.…

We prove that a $C^{\infty}$-generic area-preserving diffeomorphism of a closed, oriented surface admits a sequence of equidistributed periodic orbits. This is a quantitative refinement of the recently established generic density theorem…

For $m,n$ coprime we introduce a new statistic skip on $(m,n)$-rational Dyck paths and give a fast way to compute dinv and skip statistics. We also introduce $(m,n)$-rank words, which are in one-to-one correspondence with $(m,n)$-Dyck…

In this paper we study a subfamily of a classic lattice path, the \emph{Dyck paths}, called \emph{restricted $d$-Dyck} paths, in short $d$-Dyck. A valley of a Dyck path $P$ is a local minimum of $P$; if the difference between the heights of…

The $k$-coprime graph of order $n$ is the graph with vertex set $\{k, k+1, \ldots, k+n-1\}$ in which two vertices are adjacent if and only if they are coprime. We characterize Hamiltonian $k$-coprime graphs. As a particular case, two…

In this paper, we prove the main conjecture on $g$-areas that was announced by the first author in 2004. It states that the $g$-area of any Hamiltonian diffeomorphism $\phi$ is equal to the positive Hofer distance between $\phi$ and the…

We present a combinatorial model of configuration spaces and polytopes associated to the quotients of $\mathbb{C} A_n$, the path algebra of the linearly oriented $A_n$ quiver, i.e. the algebra of upper triangular matrices. These quotient…

In this paper we study the number of humps (peaks) in Dyck, Motzkin and Schr\"{o}der paths. Recently A. Regev noticed that the number of peaks in all Dyck paths of order $n$ is one half of the number of super Dyck paths of order $n$. He…

We discuss the combinatorics of the decorated Dyck paths appearing in the Delta conjecture framework of Haglund, Remmel and Wilson (2015) and Zabrocki (2016), by introducing two new statistics, bounce and bounce'. We then provide plethystic…

In this paper, we formulate a rational analog of the fall Delta theorem and the Delta square conjecture. We find a new dinv statistic on fall-decorated paths on a $(m+k) \times (n+k)$ rectangle that simultaneously extends the previously…

We are concerned with two separation theorems about analytic sets by Dyck and Preiss, the former involves the positively-defined subsets of the Cantor space and the latter the Borel-convex subsets of finite dimensional Banach spaces. We…

There is a natural bijection between Dyck paths and basis diagrams of the Temperley-Lieb algebra defined via tiling. Overhang paths are certain generalisations of Dyck paths allowing more general steps but restricted to a rectangle in the…

We generalize the shuffle theorem and its $(km,kn)$ version, as conjectured by Haglund et al. and Bergeron et al., and proven by Carlsson and Mellit, and Mellit, respectively. In our version the $(km,kn)$ Dyck paths on the combinatorial…

A \emph{Dyck path} is a lattice path in the first quadrant of the $xy$-plane that starts at the origin, ends on the $x$-axis, and consists of the same number of North-East steps $U$ and South-East steps $D$. A \emph{valley} is a subpath of…

The classical Chung-Feller theorem [2] tells us that the number of Dyck paths of length $n$ with $m$ flaws is the $n$-th Catalan number and independent on $m$. In this paper, we consider the refinements of Dyck paths with flaws by four…

The famous K\H{o}nig-Egerv\'ary theorem is equivalent to the statement that the matching number equals the vertex cover number for every induced subgraph of some graph if and only if that graph is bipartite. Inspired by this result, we…

It is known that the Hilbert space dimensionality for quasiparticles in an SU(2)_k Chern-Simons-Witten theory is given by the number of directed paths in certain Bratteli diagrams. We present an explicit formula for these numbers for…

We introduce an equivalence relation on the set of Dyck paths and some operations on them. We determine a formula for the cardinality of those equivalence classes and use this information to obtain a combinatorial formula for the number of…

We introduce a new statistic, skip, on rational $(3,n)$-Dyck paths and define a marked rank word for each path when $n$ is not a multiple of 3. If a triple of valid statistics (area,skip,dinv) are given, we have an algorithm to construct…