A family of \((n-1)!\) diagonal polynomial orders of \(N^ n\)
type: Item
instance of: scholarly article
title: A family of \((n-1)!\) diagonal polynomial orders of \(N^ n\)
zbMATH Open document ID: 0834.06003
DOI: 10.1007/BF01108626
published in: Order
publication date: 1996-03-31T00:00:00Z
review text: 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].
reviewed by: Konrad Engel
Mathematics Subject Classification ID: 06A06
Mathematics Subject Classification ID: 03E05
zbMATH DE Number: 828189
zbMATH Keywords: Cantor polynomial
zbMATH Keywords: interval partition
zbMATH Keywords: diagonal polynomial order
author: Adolfo Sánchez-Flores
MaRDI profile type: Publication
cites work: Diagonal polynomials for small dimensions
cites work: Polynomial indexing of integer lattice-points. I. General conceopts and quadratic polynomials
cites work: An enlarged family of packing polynomials on multidimensional lattices
cites work: Q5734593
cites work: Q5786041
description: scientific article; zbMATH DE number 828189