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Diagonal polynomials and diagonal orders on multidimensional lattices

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Abstract

HereR andN denote the real numbers and the nonnegative integers, respectively. Alsos(x)=x 1+···+x n whenx=(x 1, …,x n) inR n. A mapf:R n →R is call adiagonal function of dimensionn iff|N n is a bijection ontoN and, for allx, y inN n, f(x)<f(y) whens(x)<s(y). Morales and Lew [6] constructed 2n−2 inequivalent diagonal polynomial functions of dimensionn for eachn>1. Here we use new combinatorial ideas to show that numberd n of such functions is much greater than 2n−2 forn>3. These combinatorial ideas also give an inductive procedure to constructd n+1 diagonal orderings of {1, …,n}.

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  1. IIMAS, Universidad Nacional Autónoma de México, Apdo. Postal 70-221, 04510, México, D.F., Mexico

    L. B. Morales

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  1. L. B. Morales
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Morales, L.B. Diagonal polynomials and diagonal orders on multidimensional lattices. Theory of Computing Systems 30, 367–382 (1997). https://doi.org/10.1007/BF02679466

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