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Displaying 1081 – 1100 of 1341

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On the positivity of the number of t-core partitions

Ken Ono (1994)

Acta Arithmetica

A partition of a positive integer n is a nonincreasing sequence of positive integers with sum n . Here we define a special class of partitions. 1. Let t 1 be a positive integer. Any partition of n whose Ferrers graph have no hook numbers divisible by t is known as a t- core partitionof n . The hooks are important in the representation theory of finite symmetric groups and the theory of cranks associated with Ramanujan’s congruences for the ordinary partition function [3, 4, 6]. If t 1 and n 0 , then we define...

On the q -Pell sequences and sums of tails

Alexander E. Patkowski (2017)

Czechoslovak Mathematical Journal

We examine the q -Pell sequences and their applications to weighted partition theorems and values of L -functions. We also put them into perspective with sums of tails. It is shown that there is a deeper structure between two-variable generalizations of Rogers-Ramanujan identities and sums of tails, by offering examples of an operator equation considered in a paper published by the present author. The paper starts with the classical example offered by Ramanujan and studied by previous authors noted...

On the rainbow connection of Cartesian products and their subgraphs

Sandi Klavžar, Gašper Mekiš (2012)

Discussiones Mathematicae Graph Theory

Rainbow connection number of Cartesian products and their subgraphs are considered. Previously known bounds are compared and non-existence of such bounds for subgraphs of products are discussed. It is shown that the rainbow connection number of an isometric subgraph of a hypercube is bounded above by the rainbow connection number of the hypercube. Isometric subgraphs of hypercubes with the rainbow connection number as small as possible compared to the rainbow connection of the hypercube are constructed....

On the Rainbow Vertex-Connection

Xueliang Li, Yongtang Shi (2013)

Discussiones Mathematicae Graph Theory

A vertex-colored graph is rainbow vertex-connected if any two vertices are connected by a path whose internal vertices have distinct colors. The rainbow vertex-connection of a connected graph G, denoted by rvc(G), is the smallest number of colors that are needed in order to make G rainbow vertexconnected. It was proved that if G is a graph of order n with minimum degree δ, then rvc(G) < 11n/δ. In this paper, we show that rvc(G) ≤ 3n/(δ+1)+5 for [xxx] and n ≥ 290, while rvc(G) ≤ 4n/(δ + 1) + 5...

On the rank of random subsets of finite affine geometry

Wojciech Kordecki (2000)

Discussiones Mathematicae Graph Theory

The aim of the paper is to give an effective formula for the calculation of the probability that a random subset of an affine geometry AG(r-1,q) has rank r. Tables for the probabilities are given for small ranks. The expected time to the first moment at which a random subset of an affine geometry achieves the rank r is derived.

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