Combinatorial Inequalities Research Articles

A certain combinatorial inequality

A certain combinatorial inequality

Operator norms, multiplicativity factors, and C-numerical radii

Let V be a normed vector space over C, let B ( V) denote the algebra of linear bounded operators on V, and let N be an arbitrary seminorm or norm on B ( V). In this paper we discuss multiplicativity factors for N, i.e., constants μ>0 for which N μ ≡ μN is submultiplicative. We find that, while in the finite dimensional case nontrivial indefinite seminorms have no multiplicativity factors and norms do have multiplicativity factors, in the infinite dimensional case N may or may not have such factors. Our results are then applied in order to compute multiplicativity factors for certain generalizations of the classical numerical radius, called C-numerical radii. This is done with the help of a combinatorial inequality which seems to be of independent interest.

(1,k)-configurations and facets for packing problems

A new class of facets for knapsack polytopes is obtained. This class of inequalities is shown to define a polytope with zero–one vertices only. A combinatorial inequality is obtained from Fulkerson's max—max inequality.

Spearman's Footrule as a Measure of Disarray

Summary Spearman's measure of disarray D is the sum of the absolute values of the difference between the ranks. We treat D as a metric on the set of permutations. The limiting mean, variance and normality are established. D is shown to be related to the metric I arising from Kendall's τ through the combinatorial inequality I ≤ D ≤ 2I.

Combinatorial inequalities for semimodular lattices of breadth two

Combinatorial inequalities for semimodular lattices of breadth two

Quantitative axioms for finite affine planes

All axiom systems are derived which define finite affine planes and consist of certain combinatorial inequalities or equalities involving the number of lines connecting two distinct points the number of lines through a point the number of points on a line, the total number of points or the total number of lines. The result given in Dembowski book Finite Geometries are corrected.

Combinatorial inequalities and smoothness of functions

Combinatorial inequalities and smoothness of functions

Local matching in the function space of a partial order

An inner product space is defined on a finite partial order. Some combinatorial inequalities pertaining to chains and antichains are thereby subsumed into the theory of convex cones. In particular, a local matching condition is found to determine the cone dual to the convex cone generated by characteristic functions of maximal chains.

Operator theoretic invariants and the enumeration theory of Pólya and de Bruijn

It has been shown by M. Marcus and others that, in regard to combinatorial matrix functions and combinatorial inequalities, it is frequently fruitful to pass immediately from the consideration of permutations to the consideration of their tensor representations. Such an approach embeds the combinatorial arguments into the framework of linear algebra and frequently results in deeper theorems. It is interesting to note that certain basic combinatorial identities concerned with pattern enumeration and combinatorial generating functions can also be put into this framework. In this paper we consider one possible way of doing this.

On a class of combinatorial inequalities

In this paper we derive upper and lower bounds for combinatorial matrix functions associated with a group G in terms of the matrix functions associated with the sub-group of G.

Generalizations of some combinatorial inequalities of H. J. Ryser

Generalizations of some combinatorial inequalities of H. J. Ryser