We remark that a Golomb ruler of order n is defined to be a sequence of n distinct positive integers such that all of the entries in its difference triangle are distinct. In terms of its difference triangle, the integers in each row are distinct. For example, for arr 1,2,3, the following are. If we think of a permutation matrix as a configuration of points in the Euclidean plane at the integral positions ( i, π i) for i = 1,…, n, then for a Costas permutation, no two of the line segments determined by these points have both the same length and the same slope, and thus all the line segments they determine are distinct. A permutation of an array of integers is an arrangement of its members into a sequence or linear order. Since π is a permutation, its zeroth order differences are distinct since π has only one ( n − 1)st difference, no restriction is placed on ( n − 1)st order differences. One possible matrix is m1 in the picture.

A Costas permutation (or Costas array or Costas permutation matrix) is a permutation π of order n such that for each k = 0,1,2,…, n − 1, its k th order differences in row k of its difference triangle are distinct. I want to permute only element of columns such that each rows contains the same letter exactly once. The difference triangle is used and analyzed extensively in the literature on Costas arrays. Every row and column therefore contains precisely a single 1 with 0s everywhere else, and every permutation corresponds to a unique permutation matrix. Permutation matrices and more general classes of (0,1)-matrices are treated in. A permutation matrix is a matrix obtained by permuting the rows of an identity matrix according to some permutation of the numbers 1 to. We refer to the book on permutations and their descents. The notion of the derivative of a permutation captures the changes in consecutive entries of a permutation π, and therefore contains information about e.g.

Permute a matrix