Improved lower bounds on extremal functions of multidimensional permutation matrices

Extremal functions of forbidden multidimensional matrices

Pattern avoidance is a central topic in graph theory and combinatorics. Pattern avoidance in matrices has applications in computer science and engineering, such as robot motion planning and VLSI circuit design. A d-dimensional zero-one matrix A avoids another d-dimensional zero-one matrix P if no submatrix of A can be transformed to P by changing some ones to zeros. A fundamental problem is to ...

Lower bounds of Copson type for the transpose of matrices on weighted sequence spaces

Bounding extremal functions of forbidden 0-1 matrices using (r, s)-formations

First, we prove tight bounds of n2 1 (t−2)! α(n)t−2±O(α(n)t−3) on the extremal function of the forbidden pair of ordered sequences (123 . . . k)t and (k . . . 321)t using bounds on a class of sequences called (r, s)formations. Then, we show how an analogous method can be used to derive similar bounds on the extremal functions of forbidden pairs of 0 − 1 matrices consisting of horizontal concate...

On the Column Extremal Functions of Forbidden 0-1 Matrices

A 0-1 matrix is a matrix in which every element is either 0 or 1. The weight extremal function ex(n, P ) counts the maximum number of 1’s in an n × n matrix which avoids a pattern matrix P . The column extremal function exk(m,P ) counts the maximum number of columns that a matrix with m rows and k 1’s per column can contain such that the matrix avoids P . Set weight and column extremal function...

Generalized matrix functions, determinant and permanent

In this paper, using permutation matrices or symmetric matrices, necessary and sufficient conditions are given for a generalized matrix function to be the determinant or the permanent. We prove that a generalized matrix function is the determinant or the permanent if and only if it preserves the product of symmetric permutation matrices. Also we show that a generalized matrix function is the de...