Combinatorial representations

  title={Combinatorial representations},
  author={Peter J. Cameron and Maximilien Gadouleau and S{\o}ren Riis},
  journal={J. Comb. Theory, Ser. A},
This paper introduces combinatorial representations, which generalise the notion of linear representations of matroids. We show that any family of subsets of the same cardinality has a combinatorial representation via matrices. We then prove that any graph is representable over all alphabets of size larger than some number depending on the graph. We also provide a characterisation of families representable over a given alphabet. Then, we associate a rank function and a closure operator to any… Expand
4 Citations
Closure Solvability for Network Coding and Secret Sharing
  • M. Gadouleau
  • Mathematics, Computer Science
  • IEEE Transactions on Information Theory
  • 2013
A closure operator on a digraph closely related to the network coding instance is defined and it is shown that the constraints for network coding can all be expressed according to that closure operator, which yields bounds on the amount of information that can be transmitted through a network. Expand
Computing without memory
The famous algorithm for swapping the contents of two variables without using a buffer is generalised and it is proved that any function of all the variables can be computed this way in a number of updates which grows linearly with the number of variables. Expand
Memoryless computation: New results, constructions, and extensions
It is shown that combining variables, instead of simply moving them around, not only allows for memoryless programs, but also yields shorter programs and allows us to use only binary instructions, which is not sufficient in general when no memory is used. Expand
Decision systems in rough set theory: A set operatorial perspective
In rough set theory (RST), the notion of decision table plays a fundamental role. In this paper, we develop a purely mathematical investigation of this notion to show that several basic aspects ofExpand


Matroid representations by partitions
  • F. Matús
  • Computer Science, Mathematics
  • Discret. Math.
  • 1999
Partition representable matroids are shown to be closely related to generalized quasigroup equations read out of the matroid structure and a special morphism of partition representations, called partition isotopy, is introduced. Expand
New inequalities for subspace arrangements
  • R. Kinser
  • Mathematics, Computer Science
  • J. Comb. Theory, Ser. A
  • 2011
For each positive integer n>=4, this work gives an inequality satisfied by rank functions of arrangements of arrangement of n subspaces that can be thought of as a hierarchy of necessary conditions for a (poly)matroid to be realizable. Expand
Representation of matroids
In this paper we give a necessary and sufficient criterion for representability of a matroid over an algebraic closed field. This leads to an algorithm, based on an extension of Groebner Bases, inExpand
On secret-sharing matroids
  • P. Seymour
  • Computer Science, Mathematics
  • J. Comb. Theory, Ser. B
  • 1992
This paper shows that the Vamos matroid is not a secret-sharing matroid, by showing that for all X ⊇ E(M), the submatrix (aij : i ∈ I, j ∈ X) has precisely |S|rk(χ) distinct rows. Expand
Constructions and Uses of Pairwise Balanced Designs
A pairwise balanced design (PBD) of index unity is a pair (X,A) where X is a set (of points) and A a class of subsets A of X (called blocks) such that any pair of distinct points of X is contained inExpand
A new class of non-Shannon-type inequalities for entropies
A countable set of non-Shannon-type linear information inequalities for entropies of discrete random variables, i.e., information inequalities which cannot be reduced to the "basic" inequality I(X : Y |Z) 0. Expand
Recent Progresses in Characterising Information Inequalities
  • T. Chan
  • Mathematics, Computer Science
  • Entropy
  • 2011
A revision on some of the recent progresses made in characterising and understanding information inequalities, which are the fundamental physical laws in communications and compression, is presented. Expand
On the classification of ideal secret sharing schemes
This paper shows a relationship between ideal secret sharing schemes and matroids and shows that any subset of participants who can use their shares to determine any information about the key can in fact actually determine the key. Expand
Six New Non-Shannon Information Inequalities
Six new unconstrained non-Shannon information inequalities in four variables are given, independent of each other and of the Zhang-Yeung inequality. Expand
On a new non-Shannon-type information inequality
  • Zhen Zhang, Jun Yang
  • Mathematics
  • Proceedings IEEE International Symposium on Information Theory,
  • 2002
Makarychev, Makarychev, Romashchenko and Vereshchagin (2001) discovered a new non-Shannon-type information inequality involving 5 random variables which can be viewed as a generalization of the ZY98Expand