A family of asymptotically \(e(n-1)!\) polynomial orders of \(N^n\) (Q1568676)

Denote by \(N^n\) the \(n\)-sequences of non-negative integers. If \({\mathbf x}\in N^n\) then \(s({\mathbf x})=x_1+\cdots+x_n.\) An order of \(N^n\) is a bijection \(\alpha: N^n \rightarrow N.\) Two orders are equivalent if they differ only by a permutation of their arguments. Finally an order \(\alpha\) is diagonal if \(\alpha({\mathbf x})< \alpha({\mathbf y})\) whenever \(s({\mathbf x})< s({\mathbf y}).\) There is a long quest for larger and larger families or non-isomorphic diagonal or nondiagonal orders. This paper constructs a new family of nondiagonal orders, using an earlier diagonal order construction of Sanchez, highly improving any earlier order families.