A family of \((n-1)!\) diagonal polynomial orders of \(N^ n\) (Q1904395)

Let \(N\) be the set of nonnegative integers and let be \(n\in N\setminus \{0\}\). For \(x= (x_1, \dots, x_n)\in N^n\), let be \(s(x)= x_1+ \dots+ x_n\). A diagonal polynomial order in \(N^n\) is a bijective polynomial \(p: N^n\to N\) with real coefficients such that, for all \(x, y\in N^n\), \(p(x)< p(y)\) whenever \(s(x)< s(y)\). The Cantor polynomial \(f(x, y)= y+ (x+ y) (x+ y+ 1)/2\) is a well-known example for \(n=2\). It shows that \(N\) and \(N^2\) are equipotent. Two such orders are equivalent if a relabeling of the variables makes them identical. The author presents, for each \(n>0\), \((n-1)!\) diagonal polynomial orders which are pairwise inequivalent. This family contains also those \(2^{n- 2}\) orders which were constructed by \textit{L. B. Morales} and \textit{J. S. Lew} [``An enlarged family of packing functions on multidimensional lattices'', IBM Research Report RC 18291, to appear in Math. Syst. Theory].