AN ENLARGED FAMILY OF PACKING POLYNOMIALS ON MULTIDIMENSIONAL LATTICES

Here N = {0, 1,2,...}, while a function f on N-m or a larger domain is a packing function if its restriction f/N-m is a bijection onto N. (P acking functions generalize Cantor's [1] pairing polynomials, and yiel d multidimensional-array storage schemes.) We call two functions equiv alent if permuting arguments makes them equal. Also s(x) = x(l) +...x(m) when x = (x(l)..., x(m)); and such an f is a diagonal mapping if f(x) < f(y) whenever x, y is an element of N-m and s(x) < s(y). Lew [7 ] composed Skolem's [14], [15] diagonal packing polynomials (essential ly one for each m) to construct c(m) inequivalent nondiagonal packing polynomials on each N-m. For each m > 1 we now construct 2(m-2) inequi valent diagonal packing polynomials. Then, extending the tree argument s of the prior work, we obtain d(m) inequivalent nondiagonal packing p olynomials, where d(m)/c(m) --> infinity as m --> infinity. Among thes e we count the polynomials of extremal degree.