Divisor-Band Bijections and Totient Shells in the Triangular-Fractional Grid

We define the triangular-fractional grid\[ a(n,k)=\frac{n^2-n+k}{n}=n-1+\frac{k}{n}, \qquad n\geq 1,\quad 1\leq k\leq n,\]and study the reduced rational entries generated by this elementary chamber structure. The main result is an exact divisor-band bijection: a positive rational $r/s$ in lowest terms occurs in chamber $n$ if and only if\[ s\mid n,\qquad s(n-1)

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Pure mathematics, Mathematics/statistics & numerical data, Discrete mathematics, Applied mathematics, Mathematical analysis, Mathematics/education, Mathematics/standards, Mathematics/methods, Mathematical logic, FOS: Mathematics, Mathematics/classification, Mathematics/statistics & numerical data, Mathematics, Mathematics/history

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