Some number-theoretic combinatorial problems (Q1048581)

From the text: ``The author considers a number of number-theoretic combinatorial problems arising in the study of problems of graph theory and of the theory of error-correcting codes. He obtain simple expressions for a number of generating functions and, from them, derives consequences required for solving original problems. Suppose that \(\mathbb F_q\) is a finite field of \(q\) elements and \(\mathbb F_q[x]\) is the ring of polynomials over the field \(\mathbb F_q\). Further, we only consider normalized polynomials in \(\mathbb F_q[x]\). The Möbius function \(\mu(g)\) given on the elements of the ring \(\mathbb F_q[x]\), is defined by the canonical method as \[ \mu(g)=\begin{cases} 1 &\text{ for } g=1,\\ (-1)^r &\text{ for } g=v_1v_2\cdots v_r,\\ 0 &\text{ if } g \text{ is divisible by a square}.\end{cases} \tag{1} \] Here the \(v_i\) are irreducible polynomials of the factorial ring \(\mathbb F_q[x]\). The Möbius function (1) is a direct analog of the classical number-theoretic Möbius function defined in the ring of integers. Consider the following generating function: \(F_q(z)=\sum_{g\in\mathbb F_q[x]} \mu(g)z^{\deg g}\). Theorem 1. The following relation holds: \(F_q(z)=1-qz\). '' A number of further interesting results are proved.