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General numeration I. Gauged schemes. (English) Zbl 0562.05010

Given a sequence of pairs \(\{(G_ i,D_ i)\}\) of natural numbers, where the \(G_ i\) are strictly increasing. Sums of the form \(\sum n_ iG_ i,\) \(0\leq n_ i\leq D_ i\) are considered here. The mapping \((n_ k,...,n_ 0)\to \sum_{i\leq k}n_ iG_ i\) from a set M of numerals, called ’Neugebauer symbols’, satisfying \(0\leq n_ i\leq D_ i\) into the set W of all non-negative integers. In M, initial zeros are suppressed and M is ordered in the usual numerical order. Such an A is called a gauged scheme. Basic questions posed and answered in part, are: how does the structure of \(\{(G_ i,D_ i)\}\) affect A, especially as regards injectivity, surjectivity, preservation of order, additivity especially carrying, when the termwise sum of two numerals in M falls outside M. The most important conditions involve comparison of \(G_ k\) with \(T_ k\equiv 1+\sum_{i<k}D_ iD_ i\). The condition that for all k, \(G_ k\geq T_ k\Rightarrow\) injectivity of A, it is implied by the condition that the addition of two summands involves nothing more complicated than carrying a one ino the next place to the left, and it is equivalent to strict order preservation \((m>n\Rightarrow m>An)\), the condition that for all k, \(G_ k\leq T_ k\) is equivalent to surjectivity of A; the condition that for all k, \(G_ k=T_ k\) is equivalent to bijectivity of A. Proofs are combinatorial.
Reviewer: A.D.Wadhwa

MSC:

05A99 Enumerative combinatorics
05A15 Exact enumeration problems, generating functions

Keywords:

gauged scheme
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