Related papers: Multicolour chain avoidance in the boolean lattice
We enumerate permutations that avoid all but one of the $k$ patterns of length $k$ starting with a monotone increasing subsequence of length $k-1$. We compare the size of such permutation classes to the size of the class of permutations…
We consider the maximum chromatic number of hypergraphs consisting of cliques that have pairwise small intersections. Designs of the appropriate parameters produce optimal constructions, but these are known to exist only when the number of…
We study the generalisation of Baxter's three-colour problem to a random lattice. Rephrasing the problem as a matrix model problem we discuss the analyticity structure and the critical behaviour of the resulting matrix model. Based on a set…
For a finite poset $P$, we study the expected size of the intersection of two independent uniformly random antichains. Equivalently, we evaluate the sum of $|A\cap A'|$ over all ordered pairs of antichains. For general posets this statistic…
We study the problem of counting alternating permutations avoiding collections of permutation patterns including 132. We construct a bijection between the set S_n(132) of 132-avoiding permutations and the set A_{2n + 1}(132) of alternating,…
In this paper, we consider the problem of identifying patterns of interest in colored strings. A colored string is a string where each position is assigned one of a finite set of colors. Our task is to find substrings of the colored string…
We classify all one-class genera of admissible lattice chains of length at least 2 in hermitian spaces over number fields.
A lattice of integers is the collection of all linear combinations of a set of vectors for which all entries of the vectors are integers and all coefficients in the linear combinations are also integers. Lattice reduction refers to the…
Let $P$ be a set of $2n$ points in convex position, such that $n$ points are colored red and $n$ points are colored blue. A non-crossing alternating path on $P$ of length $\ell$ is a sequence $p_1, \dots, p_\ell$ of $\ell$ points from $P$…
There are several versions of permutation pattern avoidance that have arisen in the literature, and some known examples of two different types of pattern avoidance coinciding. In this paper, we examine barred patterns and vincular patterns.…
We give a characterization of finite sets of triples of elements (e.g., positive integers) that can be colored with two colors such that for every element $i$ in each color class there exists a triple which does not contain $i$. We give a…
The status of our understanding of the mechanisms of color confinement is reviewed, in particular the results of numerical simulations on the lattice.
It is known that a (concept) lattice contains an n-dimensional Boolean suborder if and only if the context contains an n-dimensional contra-nominal scale as subcontext. In this work, we investigate more closely the interplay between the…
A measure theoretic approach of the problem that there exits a finite unit-distance graphs in the plane that are not five (or four) colorable.
We study compositions whose parts are colored by subsequences of the Fibonacci numbers. We give explicit bijections between Fibonacci colored compositions and several combinatorial objects, including certain restricted ternary and…
In this note we introduce a family of polynomials on a matroid derived from chain Tutte polynomials which generalize the classic and ubiquitous characteristic polynomial. We show that the coefficients of these polynomials alternate and…
A modular or distributive lattice is `diamond-colored' if its order diagram edges are colored in such a way that, within any diamond of edges, parallel edges have the same color. Such lattices arise naturally in combinatorial representation…
A general method is proposed for predicting the asymptotic percolation threshold of networks with bottlenecks, in the limit that the sub-net mesh size goes to zero. The validity of this method is tested for bond percolation on filled…
An antichain of subsets is a set of subsets such that no subset in the antichain is a proper subset of any other subset in the antichain. The Dedekind number counts the total number of antichains of subsets of an n-element set. This paper…
We study pattern avoidance by combinatorial objects other than permutations, namely by ordered partitions of an integer and by permutations of a multiset. In the former case we determine the generating function explicitly, for integer…