Related papers: 1234-avoiding permutations and Dyck paths

Given two positive integers $n\geq 3$ and $t\leq n$, the permutations $\sigma,\pi \in \operatorname{Sym}(n)$ are $t$-setwise intersecting if they agree (setwise) on a $t$-subset of $\{1,2,\ldots,n\}$. A family $\mathcal{F} \subset…

We consider the problem of finding the matching map between two sets of $d$ dimensional vectors from noisy observations, where the second set contains outliers. The matching map is then an injection, which can be consistently estimated only…

We extend the path-integral formulation of Poisson-Lie duality found by Tyurin and von Unge to N=1 supersymmetric sigma-models. Using an explicit representation of the generators of the Drinfel'd double corresponding to GxU(1)^dimG we…

We determine the asymptotics of the largest cardinality of a set of Hamilton paths in the complete graph with vertex set [n] under the condition that for any two of the paths in the family there is a subpath of length k entirely contained…

Inspired by work of McKay, Morse, and Wilf, we give an exact count of the involutions in S_n which contain a given permutation \tau in S_k as a subsequence; this number depends on the patterns of the first j values of \tau for 1<=j<=k. We…

Let $\mathscr J$ be the set of inner functions whose derivatives lie in Nevanlinna class. In this note, we show that the natural map $F \to \text{Inn}(F'): \mathscr J/\text{Aut}(\mathbb{D}) \to \text{Inn}/S^1$ is is injective but not…

We introduce Mayer path homology, a new homology theory for directed path complexes obtained by equipping path complexes with an $N$-nilpotent differential. The main novelty of this work is the introduction of an $N$-differential on path…

In the Disjoint Paths problem, one is given a graph with a set of $k$ vertex pairs $(s_i,t_i)$ and the task is to connect each $s_i$ to $t_i$ with a path, so that the $k$ paths are pairwise disjoint. In the optimization variant, Max…

We investigate {\it Gottlieb map}s, which are maps $f:E\to B$ that induce the maps between the Gottlieb groups $\pi_n (f)|_{G_n(E)}:G_n(E)\to G_n(B)$ for all $n$, from a rational homotopy theory point of view.We will define the obstruction…

The action of the symmetric group $S_n$ on the set $Park_n$ of parking functions of size $n$ has received a great deal of attention in algebraic combinatorics. We prove that the action of $S_n$ on $Park_n$ extends to an action of $S_{n+1}$.…

For an integer $m \geq 2$, let $\mathcal{P}_m$ be the partition of the unit interval $I$ into $m$ equal subintervals, and let $\mathcal{F}_m$ be the class of piecewise linear maps on $I$ with constant slope $\pm m$ on each element of…

We characterize (in almost all cases) the holomorphic self-maps of the unit disk that intertwine two given linear fractional self-maps of the disk. The proofs are based on iteration and a careful analysis of the Denjoy-Wolff points. In…

A unit spherical Euclidean distance matrix (EDM) D is a matrix whose entries can be realized as the interpoint (squared) Euclidean distances of n points on a unit sphere. In this paper, given such a D and 1 \leq k < l \leq n, we present a…

We introduce the stack-sorting map $\text{SC}_\sigma$ that sorts, in a right-greedy manner, an input permutation through a stack that avoids some vincular pattern $\sigma$. The stack-sorting maps of Cerbai et al. in which the stack avoids a…

We present some combinatorial interpretations for coefficients appearing in series partitioning the permutations avoiding 132 along marked mesh patterns. We identify for patterns in which only one parameter is non zero the combinatorial…

We study graph classes modeled by families of non-crossing (NC) connected sets. Two classic graph classes in this context are disk graphs and proper interval graphs. We focus on the cases when the sets are paths and the host is a tree…

We consider the situation where one is given a set S of points in the plane and a collection D of unit disks embedded in the plane. We show that finding a minimum cardinality subset of D such that any path between any two points in S is…

Let $N$ be a compact, connected, nonorientable surface of genus $g$ with $n$ boundary components with $g \geq 5$, $n \geq 0$. Let $\mathcal{T}(N)$ be the two-sided curve complex of $N$. If $\lambda :\mathcal{T}(N) \rightarrow…

Let $S_{n}$ denote the set of permutations of $[n]=\{1,2,\dots, n\}$. For each integer $k\geq 1$, let $S_{n,k}$ be the set of all permutations of $[n]$ with exactly $k$ disjoint cycles. A subset $H\subseteq S_{n,k}$ is to be a matching if…

The exact path length problem is to determine if there is a path of a given fixed cost between two vertices. This paper focuses on the exact path problem for costs $-1,0$ or $+1$ between all pairs of vertices in an edge-weighted digraph.…