On the Number of Latin Rectangles and Chromatic Polynomial of L(Kr, s)

Using Mobius inversion formula it is shown that the total number of Latin rectangles of a given order can be expressed in terms of Mobius function for the lattice of partitions of a set and the number of colourings of certain graphs. We prove the result in a very general form. In fact, we generalize the notions of Latin rectangles and colourings of graphs and prove a theorem in this general setting. An equivalent form of the theorem which is handy for calculation is given. Various special cases are considered. In particular, we obtain the chromatic polynomials of the line graphs of K 3, k and K 4, k or equivalently the total number of 3 × k and 4 × k Latin rectangles with entries from an n -set.