A set \(S\) is called \(k\)-multiple-free if \(S \cap kS = \emptyset\), where \(kS = \{ks : s \in S\}\). Let \(N_n = \{1, 2, \ldots, n\}\). A \(k\)-multiple-free set \(M\) is maximal in \(N_n\) if for any \(k\)-multiple-free set \(A\), \(M \subseteq A \subseteq N_n\) implies \(M = A\). Let

\[A(n, k) = \{|M| : M \subseteq N_n is maximal k -multiple-free\}\].

Formulae of \(\lambda(n,k)= \max \Lambda(n, k)\) and \(\mu(n, k) = \min \Lambda(n, k)\) are given. Also, the condition for \(\mu(n, k) = \Lambda(n, k)\) is characterized.

We enumerate various families of planar lattice paths consisting of unit steps in directions \( {N}\), \({S}\), \({E}\), or \({W}\), which do not cross the \(x\)-axis or both \(x\)- and \(y\)-axes. The proofs are purely combinatorial throughout, using either reflections or bijections between these \({NSEW}\)-paths and linear \({NS}\)-paths. We also consider other dimension-changing bijections.