Alon, Noga Kupavskii, Andrey
Published in
Journal of Combinatorial Theory, Series A

A faithful (unit) distance graph in Rd is a graph whose set of vertices is a finite subset of the d-dimensional Euclidean space, where two vertices are adjacent if and only if the Euclidean distance between them is exactly 1. A (unit) distance graph in Rd is any subgraph of such a graph.In the first part of the paper we focus on the differences bet...

Bessenrodt, Christine Holm, Thorsten Jørgensen, Peter
Published in
Journal of Combinatorial Theory, Series A

Frieze patterns (in the sense of Conway and Coxeter) are in close connection to triangulations of polygons. Broline, Crowe and Isaacs have assigned a symmetric matrix to each polygon triangulation and computed the determinant. In this paper we consider d-angulations of polygons and generalize the combinatorial algorithm for computing the entries in...

Coskun, Izzet
Published in
Israel Journal of Mathematics

A Schubert class σ in the cohomology of a homogeneous variety X is called rigid if the only projective subvarieties of X representing σ are Schubert varieties. A Schubert class σ is called multi rigid if the only projective subvarieties representing positive integral multiples of σ are unions of Schubert varieties. In this paper, we discuss the rig...

Aval, Jean-Christophe DʼAdderio, Michele Dukes, Mark Hicks, Angela Le Borgne, Yvan
Published in
Journal of Combinatorial Theory, Series A

We study the statistics area, bounce and dinv on the set of parallelogram polyominoes having a rectangular m times n bounding box. We show that the bi-statistics (area,bounce) and (area,dinv) give rise to the same q,t-analogue of Narayana numbers which was introduced by two of the authors in [4]. We prove the main conjectures of that paper: the q,t...

Braun, Michael Kohnert, Axel Östergård, Patric R.J. Wassermann, Alfred
Published in
Journal of Combinatorial Theory, Series A

A t-(n,k,λ;q)-design is a set of k-dimensional subspaces, called blocks, of an n-dimensional vector space V over the finite field with q elements such that each t-dimensional subspace is contained in exactly λ blocks. A partition of the complete set of k-dimensional subspaces of V into disjoint t-(n,k,λ;q) designs is called a large set of t-designs...

Füredi, Zoltán Jiang, Tao
Published in
Journal of Combinatorial Theory, Series A

A k-uniform linear cycle of length ℓ, denoted by Cℓ(k), is a cyclic list of k-sets A1,…,Aℓ such that consecutive sets intersect in exactly one element and nonconsecutive sets are disjoint. For all k⩾5 and ℓ⩾3 and sufficiently large n we determine the largest size of a k-uniform set family on [n] not containing a linear cycle of length ℓ. For odd ℓ=...

Liu, Ricky Ini
Published in
Journal of Combinatorial Theory, Series A

Erman, Smith, and Várilly-Alvarado (2011) showed that the expected number of doubly monic Laurent polynomials f(z)=z−m+a−m+1z−m+1+⋯+an−1zn−1+zn whose first m+n−1 powers have vanishing constant term is the Eulerian number 〈m+n−1m−1〉, as well as a more refined result about sparse Laurent polynomials. We give an alternate proof of these results using ...

Kaiser, Tomáš Stehlík, Matěj Škrekovski, Riste
Published in
Journal of Combinatorial Theory, Series A

Motivated by questions about square-free monomial ideals in polynomial rings, in 2010 Francisco et al. conjectured that for every positive integer k and every k-critical (i.e., critically k-chromatic) graph, there is a set of vertices whose replication produces a (k+1)-critical graph. (The replication of a set W of vertices of a graph is the operat...

Cesaratto, Eda Matera, Guillermo Pérez, Mariana Privitelli, Melina
Published in
Journal of Combinatorial Theory, Series A

We obtain an estimate on the average cardinality of the value set of any family of monic polynomials of Fq[T] of degree d for which s consecutive coefficients ad−1,…,ad−s are fixed. Our estimate holds without restrictions on the characteristic of Fq and asserts that V(d,s,a)=μdq+O(1), where V(d,s,a) is such an average cardinality, μd:=∑r=1d(−1)r−1/...

Barrese, Kenneth Loehr, Nicholas Remmel, Jeffrey Sagan, Bruce E.
Published in
Journal of Combinatorial Theory, Series A

Goldman, Joichi, and White proved a beautiful theorem showing that the falling factorial generating function for the rook numbers of a Ferrers board factors over the integers. Briggs and Remmel studied an analogue of rook placements where rows are replaced by sets of m rows called levels. They proved a version of the factorization theorem in that s...