Related papers: Schroder combinatorics and $\nu$-associahedra
The set of Schr\"oder words (Schr\"oder language) is endowed with a natural partial order, which can be conveniently described by interpreting Schr\"oder words as lattice paths. The resulting poset is called the Schr\"oder pattern poset. We…
In this paper, we investigate the weighted Catalan, Motzkin and Schr\"oder numbers together with the corresponding weighted paths. The relation between these numbers is illustrated by three equations, which also lead to some known and new…
We present several bijections, in terms of combinatorial objects counted by the Schr\"oder numbers, that are then used (via coloring) for the construction and enumeration of rational Schr\"oder paths with integer slope, ordered rooted…
Lattice paths called $\ell$-Schr\"oder paths are introduced. They are paths on the upper half-plane consisting of $\ell+2$ types of steps: $(i,\ell-i)$ for $i=0,\ldots,\ell$, and $(1,-1)$. Those paths generalize Schr\"oder paths and some…
We study the asymptotic behavior of large data solutions to Schr\"odinger equations $i u_t + \Delta u = F(u)$ in $\R^d$, assuming globally bounded $H^1_x(\R^d)$ norm (i.e. no blowup in the energy space), in high dimensions $d \geq 5$ and…
This paper contains a partial answer to the open problem 3.11 of \cite{[H2008]}. That is to find an explicit bijection on Schr\"oder paths that inverts the statistics area and bounce. This paper started as an attempt to write the sum over…
In this paper we enumerate the number of $(k, r)$-Fuss-Schr\"{o}der paths of type $\lambda$. Y. Park and S. Kim studied small Schr\"{o}der paths with type $\lambda$. Generalizing the results to small $(k, r)$-Fuss-Schr\"{o}der paths with…
This is both an expository and research paper where we advocate a systematic study of continuous analogues of finite partially ordered sets, convex polytopes, oriented matroids, arrangements of subspaces, finite simplicial complexes, and…
We consider a sorting machine consisting of two stacks in series where the first stack has the added restriction that entries in the stack must be in decreasing order from top to bottom. The class of permutations sortable by this machine…
For any northeast path $\nu$, we define two bivariate polynomials associated with the $\nu$-associahedron: the $F$- and the $H$-triangle. We prove combinatorially that we can obtain one from the other by an invertible transformation of…
We prove that the set of matchings with a fixed number of unmatched vertices is Schur-positive with respect to the set of short chords. Two proofs are presented. The first proof applies a new combinatorial criterion for Schur-positivity,…
We discuss the combinatorics of faces of trees in the context of dendroidal sets and develop a systematic treatment of dendroidal anodyne extensions. As the main example and our motivation, we prove the pushout-product property for the…
This work is concerned with different aspects of spectrahedra and their projections, sets that are important in semidefinite optimization. We prove results on the limitations of so called Lasserre and theta body relaxation methods for…
The $n^{\text{th}}$ small Schr\"oder number is $s(n) = \sum_{k \geq 0} s(n,k)$, where $s(n,k)$ denotes the number of plane rooted trees with $n$ leaves and $k$ internal nodes that each has at least two children. In this manuscript, we focus…
We introduce and develop the theory of UMEL-shellable posets. These are posets equipped with an edge-lexicographical labeling satisfying certain uniformity and monotonicity properties. This framework encompasses classical families of…
We introduce a family of generalized Schr\"oder polynomials $S_\tau(q,t,a)$, indexed by triangular partitions $\tau$ and prove that $S_\tau(q,t,a)$ agrees with the Poincar\'e series of the triply graded Khovanov-Rozansky homology of the…
The main purpose of this paper is to show that the multiplication of a Schubert polynomial of finite type $A$ by a Schur function, which we refer to as Schubert vs. Schur problem, can be understood from the multiplication in the space of…
We prove that the computation of a combinatorial shortest path between two vertices of a graph associahedron, introduced by Carr and Devadoss, is NP-hard. This resolves an open problem raised by Cardinal. A graph associahedron is a…
Given a lattice path $\nu$, the alt $\nu$-Tamari lattice is a partial order recently introduced by Ceballos and Chenevi\`ere, which generalizes the $\nu$-Tamari lattice and the $\nu$-Dyck lattice. All these posets are defined on the set of…
We consider the nonlinear Schr\"odinger equation \begin{equation*} \Delta u = \big( 1 +\varepsilon V_1(|y|)\big)u - |u|^{p-1}u \quad \text{in} \quad \mathbb{R}^N, \quad N\ge 3, \quad p \in \left(1, \frac{N+2}{N-2}\right).\end{equation*} The…