it can be shown that the polynomials so defined satisfy the pascal identities given below, and therefore coincide with the polynomials given by the algebraic definitions. a visual way to view this definition is to associate to each word a path across a rectangular grid with sides of height and width , from the bottom left corner to the top right corner, taking a step right for each letter 0 and a step up for each letter 1. then the number of inversions of the word equals the area of the part of the rectangle that is to the bottom-right of the path.