Abstract
HereN = {0, 1, 2, ...}, while a functionf onN m or a larger domain is apacking function if its restrictionf¦N m is a bijection ontoN. (Packing functions generalize Cantor's [1]pairing polynomials, and yield multidimensional-array storage schemes.) We call two functionsequivalent if permuting arguments makes them equal. Alsos(x) =x 1 + ... +x m when x = (x 1,...,x m); and such anf is adiagonal mapping iff(x) <f(y) whenever x, y εN m ands(x) <s(y). Lew [7] composed Skolem's [14], [15] diagonal packing polynomials (essentially one for eachm) to constructc(m) inequivalent nondiagonal packing polynomials on eachN m. For eachm > 1 we now construct 2m−2 inequivalent diagonal packing polynomials. Then, extending the tree arguments of the prior work, we obtaind(m) inequivalent nondiagonal packing polynomials, whered(m)/c(m) → ∞ asm → ∞. Among these we count the polynomials of extremal degree.
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Morales, L.B., Lew, J.S. An enlarged family of packing polynomials on multidimensional lattices. Math. Systems Theory 29, 293–303 (1996). https://doi.org/10.1007/BF01201281
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DOI: https://doi.org/10.1007/BF01201281