Counting patterns in colored orthogonal arrays

Let SS be an orthogonal array OA(d,k)OA(d,k) and let cc be an rr-coloring of its ground set XX. We give a combinatorial identity which relates the number of vectors in SS with given color patterns under cc with the cardinalities of the color classes. Several applications of the identity are considered. Among them it is shown that every coloring of an orthogonal array OA(d,d−1)OA(d,d−1) contains a positive proportion of almost rainbow vectors and also of almost monochromatic vectors of every color.