Related papers: A Simple Proof Characterizing Interval Orders with…

For each prime $p$, let $I_p \subset \mathbb{Z}/p\mathbb{Z}$ denote a collection of residue classes modulo $p$ such that the cardinalities $|I_p|$ are bounded and about $1$ on average. We show that for sufficiently large $x$, the sifted set…

A balanced pair in a finite ordered set $P=(V,\leq)$ is a pair $(x,y)$ of elements of $V$ such that the proportion of linear extensions of $P$ that put $x$ before $y$ is in the real interval $[1/3, 2/3]$. We prove that every finite $N$-free…

We introduce the poset of mesh patterns, which generalises the permutation pattern poset. We fully classify the mesh patterns for which the interval [1^\emptyset,m] is non-pure, where 1^\emptyset is the unshaded singleton mesh pattern. We…

For a map $f:I \rightarrow I$, a point $x \in I$ is periodic with period $p \in \mathbb{N}$ if $f^p(x)=x$ and $f^j(x)\not=x$ for all $0<j<p$. When $f$ is continuous and $I$ is an interval, a theorem due to Sharkovskii (\cite{BC}) states…

Klavik et al. [arXiv:1207.6960] recently introduced a generalization of recognition called the bounded representation problem which we study for the classes of interval and proper interval graphs. The input gives a graph G and in addition…

A barcode is a finite multiset of intervals on the real line, $B = \{ (b_i, d_i)\}_{i=1}^n$. Barcodes are important objects in topological data analysis, where they serve as summaries of the persistent homology groups of a filtration. The…

Let A be a set of integers and let h \geq 2. For every integer n, let r_{A, h}(n) denote the number of representations of n in the form n=a_1+...+a_h, where a_1,...,a_h belong to the set A, and a_1\leq ... \leq a_h. The function r_{A,h}…

Let X be a finite set. This paper describes some topological and combinatorial properties of the poset \Omega_X of order relations on X. In particular, the homotopy type of all the intervals in \Omega_X is precisely determined, and the…

Let $(P,\leq)$ be a finite poset (partially ordered set), where $P$ has cardinality $n$. Consider linear extensions of $P$ as permutations $x_1x_2\cdots x_n$ in one-line notation. For distinct elements $x,y\in P$, we define…

The higher Bruhat order is a poset of cubical tilings of a cyclic zonotope whose covering relations are cubical flips. For a 2-dimensional zonotope, the higher Bruhat order is isomorphic to a poset on commutation classes of reduced words…

A careful study is made of embeddings of posets which have a convex range. We observe that such embeddings share nice properties with the homomorphisms of more restrictive categories; for example, we show that every order embedding between…

A natural unit interval order is a naturally labelled partially ordered set that avoids patterns ${\bf 3} + {\bf 1}$ and $\bf 2 + 2$. To each natural unit interval order one can associate a symmetric function. The Stanley-Stembridge…

We prove that partially ordered set has finite number of finite-dimensional indecomposable nonequivalent Hilbert representations with orthoscalarity condition if and anly if it has finite number of indecomposable linear representations. We…

We unify several seemingly different graph and digraph classes under one umbrella. These classes are all broadly speaking different generalizations of interval graphs, and include, in addition to interval graphs, also adjusted interval…

We study the problem of extending any order-preserving Lipschitz function that maps a subset of a partially ordered Hilbert space X into a Hadamard poset Y without increasing its Lipschitz constant and preserving its monotonicity. This sort…

For a bipartite graph $G$ with parts $X$ and $Y$, an $X$-interval coloring is a proper edge coloring of $G$ by integers such that the colors on the edges incident to any vertex in $X$ form an interval. Denote by $\chi'_{int}(G,X)$ the…

This paper studies three natural pre-orders of increasing generality on the set of all completely non-unitary partial isometries with equal defect indices. We show that the problem of determining when one partial isometry is less than…

The elements of a finite field of prime order canonically correspond to the integers in an interval. This induces an ordering on the elements of the field. Using this ordering, Kiss and Somlai recently proved interesting properties of the…

In this article we prove in the main theorem that, there is a bijection between the isomorphism classes of a certain type of real hyperplane arrangements on the one hand, and the antipodal pairs of convex cones of an associated…

We consider the problems of finding optimal identifying codes, (open) locating-dominating sets and resolving sets of an interval or a permutation graph. In these problems, one asks to find a subset of vertices, normally called a…